Fretting Fatigue Analysis and Palliatives by Gary W. Kirkpatrick B.S., Civil Engineering Case Western Reserve University, 1990 Submitted to the Departments of Ocean Engineering and Materials Science and Engineering in partial fulfillment of the requirements for the degrees of Naval Engineer and Master of Science in Materials Science and Engineering at the Massachusetts Institute of Technology June 1999 @ 1999 Massachusetts Institute of Technology 1999. All rights reserved.
Signature of Author ...... Departments of Ocean Engineering and Materials Science and Engineering May 7, 1999
Certified by . Subra Suresh Richard P. Simmons Professor of Metallurgy Professor of Materials Science and Engineering Thesis Supervisor
Certified by ...... - Koichi Masubuchi Professor of Ocean Engineering and Materials Science Thesis Reader
Accepted by ...... ------...... Linn Hobbs John F. Elliott Professor of Materials Chairman, DepartmentA Conimittee on Graduate Students
Accepted by ...... - . Art.... Aaggeroer
Ford Professor of Engineering Chairman, Departmental Committee on Graduate Studies MASSACHUSETTS INSTITUTE
LIBRARIES Fretting Fatigue Analysis and Palliatives by Gary W. Kirkpatrick
Submitted to the Departments of Ocean Engineering and Materials Science and Engineering on 7 May 1999, in partial fulfillment of the requirements for the degrees of Naval Engineer and Master of Science in Materials Science and Engineering
Abstract The deleterious effects of fretting on the fatigue properties of a material have been known since the work of Warlow-Davies in 1941. However, a widely accepted fretting fatigue life prediction method still does not exist and debates persist as to what the critical parameter(s) for fretting fatigue are. This work demonstrates that the surface stresses due to contact can be used to characterize the damaging effects of fretting on the fatigue properties of a material. The elastic stress analysis performed by Hamilton for a sphere on flat plane fretting geometry is used to determine the stresses due to contact for various experimental conditions provided in previous work. It is shown that the specimen fretting fatigue life is directly related to the surface stresses induced in the material by fretting. Fretting fatigue life prediction methods are analyzed. The stress-life approach, which is widely used in industry, is examined for its applicability for fretting fatigue conditions. The damage tolerant approach is used to determine the relative magnitudes of the crack nucleation and long crack propagation stages of fretting fatigue life. An analysis of fretting fatigue palliatives, focusing on shot peening and coatings, is pre- sented. A systematic method to determine the optimum shot peening depth for fretted com- ponents is proposed. The method uses the elastic stress field expressions derived by Hamilton for a sphere on flat surface contact geometry and is demonstrated for Ti-6A1-4V. However, the method is general and can be applied for any material and any contact geometry which allows analytic evaluation of the stress fields.
Thesis Supervisor: Subra Suresh Title: Richard P. Simmons Professor of Materials Science and Engineering Professor of Mechanical Engineering
2 Contents
1 Introduction 10 1.1 Definitions and Background ...... 10 1.2 Motivation and Objectives ...... 12
2 Review of Sphere on Flat Plane Contact Mechanics 15
2.1 Introduction ...... 15
2.2 Hertzian Contact ...... 15
2.2.1 Sliding Elastic Sphere on a Flat Surface .. .. . 16 2.2.2 Partial Sliding ...... 17
2.3 Hamilton's Analysis ...... 18
3 Experimental Results 24
3.1 Introduction ...... 24 .
3.2 Al 7075-T6 Experimental Results ...... 24
3.3 Ti-6A1-4V Experiments ...... 31 .
3.4 Experimental Observations ...... 33 3.4.1 Al 7075-T6 ...... 34 3.4.2 Ti-6Al-4V ...... - ...... 37
4 Fretting Fatigue Life Prediction Analysis 42
4.1 Introduction ...... 42 .
4.2 Total Life Approach ...... 44
4.3 Damage Tolerant Approach ...... 51
3 4.3.1 A l 7075-T 6 ...... 51 4.3.2 T i-6A l-4V ...... 55
5 Palliatives 58
5.1 Introduction ...... 58 5.2 Shot Peening ...... 58 5.2.1 Material Yielding Analysis ...... 63 5.2.2 Depth of Yielding ...... 65 5.2.3 Maximum Principal Stress Analysis ...... 69
5.2.4 Integrated Yielding and Maximum Principal Stress Analysis (Optimum Shot Peening Depth) ...... 70
5.3 Stress Intensity Range Threshold Analysis ...... 74 5.4 Protective Coatings ...... 74
6 Conclusions and Future Work 77
6.1 C onclusions ...... 77 6.2 Future Work ...... 78
A Ti-6Al-4V Yielding Analysis Program 80
B Depth of Yielding Analysis 85
C Maximum Principal Stress Analysis 90
D Hamilton's Expressions 94
D.1 Normal Load Stresses ...... 94
D.2 Tangential Load Stresses ...... 95
4 List of Figures
1-1 Simplified schematic of bridge type fretting pads...... 12
2-1 (a) Fretting contact between a shpere and a flat surface. (b) Top view showing circular contact region and stick zone, c...... 16
2-2 Contact stresses at the surface due to a sphere sliding on a flat surface for Al 7075-T 6...... 20
2-3 Contact stresses due to a sphere sliding on a flat surface for Al 7075-T6, z = 20 m...... 20 2-4 Plot demonstrating the effects of stick-slip conditions on u. for Al 7075-T6 (c/a = 0.5)...... 22
3-1 Simplified diagram of the fretting fatigue testing apparatus...... 25
3-2 Plot of effective stress due to contact versus number of cycles to failure for ab ~ 83 M P a...... 28
3-3 Number of cycles to failure versus Ub, S - N curve for constant contact load conditions ...... 30
3-4 Plot of the knock-down factor versus effective stress due to contact for Al 7075-T6. 32
3-5 Plot of effective stress due to contact versus number of cycles to failure for Ti-
6A-4V (Ub ~ 300 M Pa)...... 34
3-6 Picture of the left fretting pad used for test 33 with the approximate size of the stick zone shown (Al 7075-T6)...... 35
3-7 Picture of the left side of the specimen used for test 33 with the approximate size of the stick zone shown (Al 7075-T6)...... 37
5 3-8 Picture of cracking on the left fretting pad used for test 32 with the approximate
circular area of contact shown (Al 7075-T6)...... 38
3-9 Picture of the left fretting pad used for test 5 (Ti-6A1-4V) ...... 39 3-10 Picture of the left side of the specimen used for test 5 (Ti-6A1-4V)...... 40
3-11 Picture of cracking in the right fretting pad used for test 2 (Ti-6A1-4V)...... 41
4-1 Comparison of the S-N curves for plain fatigue and fretting fatigue for Al 7075-T6
(Uec = 337 M Pa)...... 43 4-2 Plot of the Fretting Correction Factor, F versus the effective stress due to contact,
b= 83 M Pa...... 47 4-3 Plot of F versus applied bulk stress, while maintaing constant contact loads for Al 7075-T6 (acc = 337 MPa)...... 47
4-4 Plot of the combination of bulk and contact loading above which fretting fatigue failure will occur for Al 7075-T6...... 48 4-5 Revised plot of F versus oec for Al 7075-T6, Ob = 83 MPa...... 49 4-6 Plot of the predicted behavior of F versus -ec for various bulk loadings (Al 7075-T 6)...... 50 4-7 Plot of F versus -ec for Ti-6A1-4V (Ub ~ 300 MPa)...... 51
4-8 Diagram showing the elliptical crack geometry...... 53
5-1 Plot showing approximate relationship between shot peened and unpeened spec-
imens for plain and fretting fatigue...... 59
5-2 Typical stress profile after shot peening and the assumed stress profile used in this w ork...... 61
5-3 Residual stress equilibrium diagram...... 61
5-4 Diagram showing the detrimental effects of shot peening if the residual stresses are relieved [44]...... 62
5-5 The combination of contact and bulk loading which causes the total effective
stress to equal the yield stress plotted versus the coefficient of friction...... 65
5-6 Plot of yielding depth, z versus maximum contact pressure for various bulk load- ings (p = 1.0)...... 66
6 5-7 Depth of yielding versus maximum contact pressure for various bulk stresses (p = 0.8)...... 67
5-8 Depth of yielding versus maximum contact pressure for various bulk stress load- ings (p = 0.5) ...... 68
5-9 Depth of yielding versus maximum contact pressure for various bulk stress load- ings (p = 0.3) ...... 68
5-10 The depth at which the maximum principle stress is less than or equal to zero plotted versus maximum contact pressure, for magnitudes of bulk stress ([t = 1.0). 70
5-11 The depth at which the maximum principal stress is less than or equal to zero plotted versus maximum contact pressure for various magnitudes of bulk stress
(p = 0.8)...... 71 5-12 The depth at which the maximum principal stress is less than or equal to zero
plotted versus the maximum contact pressure for various magnitudes of bulk
stress (/t = 0.5)...... 71
5-13 Optimum peening depth plotted versus maximum contact pressure for At = 1.0
and Ub = 0.4oy ...... 73
5-14 Optimum peening depth plotted versus maximum contact pressure for At = 0.8,
U~b= 0-10' ...... 73 5-15 Comparison of the required peening depths for the maximum principal stress
analysis and the threshold stress intensity range analysis...... 75
7 List of Tables
2.1 Properties of 7075-T6 aluminum alloy ...... 19
3.1 Al 7075-T6 fretting fatigue experiments ...... 26
3.2 Al 7075-T6 Constant Bulk load data analysis ...... 27
3.3 Stress variation as the normal load changes for a constant maximum tangential load...... 29
3.4 Al 7075-T6 stress analysis for constant contact loading ...... 30
3.5 Data used to plot the knock-down factor curve...... 31
3.6 Ti-6A1-4V material properties...... 32
3.7 Test parameters for Ti-6A1-4V fretting fatigue tests ...... 32
3.8 Ti-6A1-4V stress analysis ...... 33
3.9 Comparison of the analytical versus actual values for the fretting geometry, Al
7075-T6 (* indicates value could not be determined)...... 36
3.10 Comparison of the analytical versus actual fretting geometry (Ti-6A-4V). .. .. 39
4.1 The necessary data to determine F and its relationship with the contact stresses. 46
4.2 Fretting correction factors for different bulk stresses, constant contact loads. .. . 46
4.3 Values used in the Paris life calculations for Al 7075-T6. Consistent units for C are m eters and M Pa...... 54 4.4 Long crack propagation life for Al 7075-T6...... 56
4.5 Values used for the propagation life calculations for Ti-6A1-4V...... 57 4.6 Values of C and m for STOA, Bi-modal Ti-6A1-4V and Bi-modal Ti-6A1-4V . . 57 (units are meters and MPa)......
8 4.7 Ti-6Al-4V long crack propagation life...... 57
5.1 Parameters and respective values for the threshold stress intensity range analysis. 74
9 Chapter 1
Introduction
1.1 Definitions and Background
Fretting fatigue generally refers to the degradation of the fatigue properties of a material due
to cyclic relative displacement between two contacting materials. The most severe impact on
the fatigue properties occurs when displacement amplitudes are 10-50 jam [21]. The fretting
action produces considerable wear damage to the material and also produces very large contact
stresses. The reduction in fatigue life under these circumstances can be quite significant. The
National Materials Advisory Board [1] gives the following definitions.
" Fretting . Fretting is a surface damage phenomenon occurring on two contacting surfaces
having oscillatory relative motion of small amplitude.
" Fretting-Initiated Fatigue . A condition where the material fatigue strength is de-
graded by the presence of the following:
1. Surface stress concentrations resulting at sites of fretting pits.
2. Surface and subsurface stresses resulting from rubbing friction contact combined
with the stresses resulting from the overall fluctuating stress field.
There are many practical examples of structures in which fretting occurs. These structures
include: bolted and riveted joints, suspension cables, metallic coil ropes, key-way shaft cou- plings, coil wedges of generator rotors and blade dovetail contact sections of turbine engines
10 [2]. The fretting problem is made worse by the fact that many of the structures that undergo fretting cannot be easily inspected. The first detection of a crack may be when the structure fails. The first reported observation of the occurrence of fretting during fatigue testing was made by Eden, Rose, and Cunningham [3] in 1911. They observed corrosion in the holders of the fatigue test specimens. The corrosion was such that they had difficulty removing the broken test specimens from the testing apparatus for the heavily loaded tests which lasted for several days. Eden, Rose, and Cunningham [3] also recorded the first attempt of the use of palliatives for fretting fatigue. They oiled the fatigue specimens before putting them in place, however they reported that this was only partially effective. The specimens were easier to remove, but corrosion products were still produced. The tests conducted by Eden et al [3] were not fretting fatigue tests, the observed fretting was noted but not analyzed.
According to Hoeppner [4] the first systematic investigation of fretting was performed by
Tomlinson in 1927, in which he determined that surface damage was related to the relative
displacement between the contacting surfaces. Warlow-Davies [5] investigated the impact of
fretting on the fatigue properties of a material in 1941 and demonstrated that fretting reduced fatigue strength. He did not perform fretting and fatigue tests simultaneously, but fretted the
specimens and then performed fatigue tests on the fretted specimens. Warlow-Davies performed
the tests on medium-carbon steel and for a nickel-chromium-molybdenum alloy steel. The loss
in fatigue strength after fretting was found to be 13 percent and 18 percent respectively. By
separating the fretting and fatigue portions of the test, Warlow-Davies was able to measure
the reduction in fatigue strength due only to the surface damage caused by fretting. Later
research, such as that conducted by Waterhouse [29], showed that simultaneous fretting and
fatigue resulted in much higher reductions in fatigue strength. There have been numerous other experiments demonstrating the deleterious effects of fretting on fatigue strength. Birch
[11], Szolwinski & Farris [30] and Adibnazari & Hoeppner [15] are just a few of the other works
that have also demonstrated the damaging effects of fretting on fatigue life.
Much work has been performed in fretting fatigue since the investigation conducted by
Warlow-Davies. This thesis will not attempt to reference all the major work performed in the area of fretting fatigue. Hoeppner in [4] provides a historical review on fretting fatigue
11 specimen
P P
Fretting pads
Figure 1-1: Simplified schematic of bridge type fretting pads. research as well as a brief review of the current state of knowledge, which is part of an entire work devoted to the topic of fretting fatigue. Birch [11] also provides a brief summary of fretting research.
1.2 Motivation and Objectives
Although the damaging effects of fretting on the fatigue properties of a material have been reported quite extensively since the work of Warlow-Davies, there is still not a consensus on the critical parameter(s) for fretting fatigue. Waterhouse [9] states that the main factor in producing fatigue failure is the additional shear stress in the contact region caused by the frictional forces between the contacting bodies, while Nishioka and Hirakawa [10] stress the importance of the relative slip amplitude and Adibnazari and Hoeppner [15] demonstrate the importance of the normal pressure on fretting fatigue life. According to Birch [11], Beard lists more than fifty parameters of concern for fretting fatigue. The lack of consensus on the critical fretting parameter(s) is likely due to the experimental configurations that have been used in the past. Many of the past fretting fatigue experiments have been performed using bridge type pads as shown in Figure 1-1. The bridge pads produce stress singularities at the contact edges making a stress analysis very difficult. Many of the experimental configurations in the past also lacked the ability to measure all of the fretting parameters, specifically the oscillating tangential contact load,
12 which is a very important fretting parameter. If the oscillating tangential load is not present, the problem becomes an indentation problem and not fretting. This inability to measure the contact load conditions again limited the information available for analysis. The first objective of this work is to determine the critical fretting fatigue parameter(s). The limitations discussed above are not in this work. The work of Birch [11] and the ongoing experimental work of Conner [12] provide the experimental data used in this work. The testing
apparatus used by Birch and Conner employed spherical fretting pads. The spherical fretting pads allow analytical expressions for the stress field due to contact to be determined, which was first done by Hamilton and Goodman [13]. The testing apparatus used by Birch and
Conner, described in detail in [25], allows measurement of all the fretting parameters required to determine the stress fields due to contact. The expressions given in [13] were expressed again later by Hamilton [14] in a more convenient explicit form. The form developed by Hamilton
[14] is implemented in this work. Hamilton and Goodman provide the elastic stress fields.
This does not present a problem in this work as the stresses remained in the elastic region for
all but a few tests. The expressions given by Hamilton [14] are for a global sliding contact, which is not the
condition for most of the experiments analyzed in this work. Most of the tests analyzed in
this thesis have slip/stick conditions. Chivers and Gordelier [16] show how the Hamilton
expressions must be modified to accommodate slip/stick fretting conditions. This modification
has been implemented in this work for the slip/stick experimental conditions when necessary.
It will be demonstrated in this work that the critical fretting parameters can be captured by
one parameter, the maximum von Mises effective stress at the surface due to contact.
The second objective of this work is to analyze fretting fatigue life prediction methods
using past experimental work. A widely accepted fretting fatigue life prediction method does
not yet exist. The stress life approach, which is widely used in industry today, is examined
for its applicability for fretted components. There have been several life prediction methods
proposed for fretting fatigue, but none have utilized the stress-life approach. Edwards [6] in
1981 proposed a fretting fatigue life prediction method using linear elastic fracture mechanics.
Faanes and Fernando [7] in 1994 also proposed a life prediction method using linear elastic
fracture mechanics, but added a short crack correction to account for the effect of the high
13 stress gradient due to fretting on short crack growth. The success of linear elastic fracture mechanics applied to fretting has been limited by the difficulty in developing stress intensity factors to be used to predict short crack growth rates. Giannakopoulos, Lindley and Suresh
[46] propose a method which circumvents this problem for sharp edged contacts in their crack analogue model. Szolwinski and Farris [30] in a recent paper propose a fretting fatigue life prediction method that uses the Smith-Watson-Topper expression [8] to predict the number of cycles to crack nucleation and employs linear elastic fracture mechanics to predict the long crack growth rate, thereby avoiding the short crack growth rate problem. This work will examine both the stress-life approach and the damage tolerant approach used to predict fatigue life.
The final portion of this work analyzes fretting fatigue palliatives; coatings and shot peening.
Shot peening has been shown by Waterhouse [38], Bignonnet [391, and Chivers and Gordelier [41] to be an effective fretting fatigue palliative. However, a detailed analysis of the shot peening process applied to fretted components with the objective of determining the optimum peening depth has not yet been performed. Fretting produces large tensile and compressive stresses at the contact edges and both must be considered when determining the optimum peening depth.
A systematic method to determine the optimum peening depth for a fretted component is given in this work, using the elastic stress field expressions provided by Hamilton. This analysis also includes the effects of changing the coefficient of friction on the optimal peening depth. The elastic stress fields due to contact will also be used to provide a brief discussion of the benefits of protective coatings for fretting fatigue.
This thesis is organized in the following manner: Chapter 2 provides a description of the sphere on flat plane contact mechanics used in this work. Chapter 3 uses the works of Hamilton
[14], and Chivers and Gordelier [16] to analyze the experimental results provided by Birch [11], and Conner [12]. Chapter 4 analyzes fretting fatigue life prediction. Chapter 5 discusses fretting fatigue palliatives and provides a method to determine the optimum peening depth for a fretted component. Chapter 6 provides the conclusions from this work and suggestions for further study in work related to fretting fatigue.
14 Chapter 2
Review of Sphere on Flat Plane Contact Mechanics
2.1 Introduction
Before the analysis of Hamilton [14] and Chivers and Gordelier [16] is applied to experimental tests, a brief review of the sphere on flat plane contact mechanics is presented. The expressions
reviewed here serve as the foundation for the analysis performed in this work. The discussion
begins with the analysis conducted by Hertz in 1882.
2.2 Hertzian Contact
Over one hundred years ago Hertz analyzed the problem of normal, frictionless contacting
spheres. Hertz solved for the contact pressure distribution, p(r) and the circular radius of
contact, a, by solving the integral equation governing the boundary conditions. The radius of
contact as determined by Hertz [17] is given by:
(2/1 1 y1 - V2 a 4(R137rPR1R2 + R2) . 7rE1 irE2
where E and v are Young' s Modulus and Poisson's ratio respectively and R is the radius of
curvature. The indices 1,2 refer to the two contacting bodies. The solution derived by Hertz
15 P
R
(a) Q R 2a x
(b) Ix
Y
Figure 2-1: (a) Fretting contact between a shpere and a flat surface. (b) Top view showing circular contact region and stick zone, c. for normal contacting spheres can be applied to a sphere on a flat plane geometry by noting
that = R e + 1 , Rpiane = 00, therefore R = Regf = Rsphere. Huber [18] in 1904, Ref f- Rshere Rplane was the first to solve for the stress fields beneath the contacting surfaces.
2.2.1 Sliding Elastic Sphere on a Flat Surface
The indentation problem solved by Hertz and Huber becomes a fretting problem when an
oscillatory tangential force is applied to the sphere. The problem geometry is shown in Figure 2-1. Once the tangential force, Q, is introduced, sliding or partial slip is also introduced. The
Amonton-Coulomb law relates the normal pressure and the tangential tractions, q by:
p P (2.2) p P
where p is the coefficient of kinetic friction.
As given first by Mindlin [19] in 1949 and again later by Hamilton [14] the stresses at the
16 contact surface for incipient sliding conditions are given by:
-3P 3 2 O-z =- q-a Va2 - r , 0 < r < a (2.3)
-3z 2 2 a -r , 0 < r < a (2.4) 27ra 3
Tyz and the boundary stresses outside the circular area of contact are zero. As mentioned above, these boundary conditions only apply under conditions of gross sliding, that is when Q = taP. The expression given by Equation 2.3 is also the expression for contact pressure due to indentation without sliding and the expression given for the contact radius by Equation 2.1 is also unchanged by the addition of a tangential load. Therefore, the presence of the shear stresses on the surface has not affected either the circular contact radius or the contact pressure.
2.2.2 Partial Sliding
When the applied tangential force is less than the product of paP, macroscopic sliding does not occur, however frictional traction at the contact interface leads to small relative displacement over part of the interface. This small relative motion is commonly referred to as micro-slip. The remaining portion of the interface deforms without any relative motion. The region in which no relative motion occurs is referred to as the stick region. All the points on the surface within the stick region undergo the same tangential displacements. The stick region is shown in Figure 2-1. Mindlin [20] has shown that a stick region exists in the contact interface and the radius, c, of the outer boundary of the stick region to be given by:
- =/- c < r < a. (2.5) a I -p
As shown in Figure 2-1, the stick region is concentric with the contact area, this geometric relationship is also observed for experimental specimens. Just as the boundary conditions had to be reestablished for sliding, they must again be modified to account for micro-slip. The stresses at the contact surface are now given by [19]:
2 2 2 2 a - r , c < r < a and (2.6) 7'x-27a2
17 2 TzX = 2 {(a2 r2)1/2 (c2 r2)1/ } r < c. (2.7)
The expression for the normal pressure is the same as in the global sliding condition. The relative tangential displacements are given by [19]:
3 _3p1P r2-v 1 2-v 2 1(Q 1 6 -+ G J{ - -- 2) . (2.8) 16a G1 G2 pP
2.3 Hamilton's Analysis
As mentioned in the previous section, the stress fields beneath the surface of a circular indenter were first derived by Huber in 1904. However, the elastic stress fields beneath the surface of a circular sliding contact on a flat surface were not derived until 1966 by Hamilton and Goodman
[13]. Mindlin's work in circular sliding contact focused primarily on surface compliance and the micro-slip problem. He did not analyze the stresses below the contact surface.
It must be stressed that the equations given by Hamilton are for a global sliding sphere
and not for stick/slip conditions. It was shown earlier that the boundary stresses are not the same for the two conditions. The stress equations for the global sliding condition must
be modified slightly to obtain the correct stresses for situations involving micro-slip. Chivers
and Gordelier [16] demonstrate how the expressions must be modified. The results of a stress
analysis performed using the expressions given by Hamilton are shown in Figure 2-2. The
expressions used to generate this plot are given in Appendix D. The material used to generate
this plot was Al 7075-T6. Some material properties for this alloy are listed in Table 2.1 [11]
[21] [22]. Spherical fretting pads with a one inch radius were used and global sliding is assumed.
Using the values for M, E, and v in Table 2.1 and the elastic stress field equations given by
Hamilton, the stresses in the x and y directions have been plotted in Figure 2-2 along with
the von Mises effective stress, Oeff, at z = 0, y = 0. The stresses have been normalized by
the maximum contact pressure, Po, Po = 3. The von Mises effective stress parameter, -eff, has been used because this is a multiaxial problem. The effective stress is used to convert the
multiaxial stress state to a single stress parameter. The expression for the effective stress is
18 Material Property Value Young's modulus (E) 72 GPa Poisson ratio (V) 0.3 Monotonic yield stress (cry) 469 MPa Coefficient of kinetic friction (p) 1.2 Tensile strength (-TS) 579 MPa Endurance limit (Jend) 179 MPa Fatigue strength coefficient (-'() 1317 MPa Fatigue strength exponent (b) -0.126 Fatigue limit stress (OD) 230 MPa Fatigue ductility coefficient (q) 0.19 Fatigue ductility exponent (c) -0.56
Table 2.1: Properties of 7075-T6 aluminum alloy given by:
2 2 2 [(-x - (-y) + (Oy -oz) + (Oz - x) +6 (T2 + T2 + Tr2 (29 Creff = /2X X Z . (2.9)
The stress fields displayed in Figure 2-2 are not the only non-zero stress fields at the surface; the shear stress Tzx and the normal stresses in the z-direction are also non-zero, however they both go to zero at the contact edges. The stresses below the surface at a depth of 20 Am is plotted in Figure 2-3. Comparing Figures 2-2 and 2-3, it can be seen that the stresses due to contact quickly decay away, especially the tensile stress at the trailing edge of contact, x = -a, in the x-direction. From Figure 2-2 it can be seen that there is a large peak in a-x at the trailing edge of contact. This stress is the highest stress on the plot. In Figure 2-3, the peak tensile stress at the trailing edge is much lower in magnitude and the slope of the a-x curve is also much smoother.
The maximum von Mises effective stress is also at the surface. This is true for a sphere sliding on a flat surface when a > 0.3. When A = 0, or when the tangential force, Q, is zero, the maximum von Mises stress shifts below the surface to the point z = 0.5a, for V = 0.3 [14].
However, for a sphere fretting on a flat plane, the maximum von Mises and the maximum tensile stresses generally occur at the surface. Although the maximum tensile stress has a severe peak at the trailing edge of contact at the surface, the von Mises effective stress is roughly constant over a very large region as can be seen in Figure 2-2.
19 3 I~~~~~~~~ ~ ~ ~ I ' ' ' ' ar eff -- a 2
1
C- 0 b -y -1
-2
. 1 i I I I I I I I I I I I I I I I I I -3 -1.5 -1 -0.5 0 0.5 1 1.5 xla
Figure 2-2: Contact stresses at the surface due to a sphere sliding on a flat surface for Al 7075-T6.
2
1.5 -~ - eff
1
0.5 0. 0 y~
-0.5 -c -1
-1.5 -1.5 -1 -0.5 0 0.5 1 1.5 x/a
Figure 2-3: Contact stresses due to a sphere sliding on a flat surface for Al 7075-T6, z = 20 pm.
20 The analysis performed in this work will focus on the surface stresses, at the trailing edge of contact, with the exception of the shot peening analysis, which analyzes below the surface and inside the contact edges. The contact stresses are maximum at the surface and the maximum tensile stress occurs in the material at the trailing edge of the contact; therefore, this work will focus its analysis on this location. The von Mises effective stress is just as high at other locations on the surface, however this is in general due to large compressive stresses at that location. It is assumed that the large tensile stress is responsible for crack nucleation. The contact stresses given by Hamilton can be greatly simplified for x = -a, z = 0 and y = 0, the problem becomes a biaxial one. The only non-zero stresses at this location are ox and oy, and
are given by: (1 - 2v) 4 + v -x= Po[ ± 7y,1 x = ~Fa, and (2.10) L 3 8 j
= P [ (2v 3- 1) 37rvtt~, xY= -Fa. (2.11)
Note carefully the position of the signs. The maximum tensile stress in both the x and y
directions occurs at the trailing edge of contact. Again, it must be emphasized that the
stresses given in equations 2.10 and 2.11 are for full sliding and not for stick/slip conditions.
Figure 2-4 displays the curves for o- under incipient sliding and stick/slip conditions. As can
be seen in this figure, when stick/slip is accounted for, o- is reduced for the given contact load
conditions. This figure was generated using a c/a ratio of 0.5.
Recall that the ratio of c/a indicates how close to full sliding or full stick the fretting
condition is. As c/a -+ 0, global sliding is introduced, when c/a = 1, full stick conditions
exist. From Figure 2-4, it can be seen that the magnitude of the maximum tensile stress
is reduced when micro-slip is taken into consideration. This fact will become more relevant
later during the shot-peening analysis of this work. The reduction in stress under stick/slip
conditions is in direct conflict with experimental results. Experiments have shown that fretting
is most damaging to the fatigue properties of a system under micro-slip conditions with relative
displacements of 10-50 pm. The introduction of global sliding improves the fretting fatigue
life. It has been postulated that this improvement under sliding conditions may be due to
the wearing away of fatigue cracks from the surface, so that they do not become propagating
fatigue cracks. This is one example of the limitations of the stress analysis.
21 2.5
2 ------
------15 ------1.5------i------0 .5 -- -- ~------P artial Slip
-0.50
-0.5 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 x/a
Figure 2-4: Plot demonstrating the effects of stick-slip conditions on cr., for Al 7075-T6 (c/a = 0.5).
In most practical structures and scientific experiments, the tangential load, Q, which changes the problem from one of indentation to fretting, is caused by an applied cyclic bulk stress.
Therefore, to perform a complete analysis, the stresses induced in the material by the bulk load
must also be accounted for. If the material behaves elastically, the principle of superposition
can be applied. Using superposition, the stresses at the surface, at r = a, are given by:
==xtot -xc + Uxb and (2.12)
Uytot = O-yc + gyb , (2.13)
where c-xtot and oytot are the total stresses in their respective directions. oxc and -yc are
the contact stresses in their respective directions and -xb, Oyb are the bulk stresses in the x and y directions respectively. The condition described above, that of a cyclic bulk load in
combination with the applied contact loads, was the method used to produce the experimental results analyzed in this work. The bulk load also produces eccentricity in the contact geometry.
The contact area and stick zone are no longer perfectly circular. This effect is discussed in
22 detail in [23].
23 Chapter 3
Experimental Results
3.1 Introduction
As shown earlier, the elastic stress fields for a sphere sliding on a flat plane are known and can be found as functions of the normal and tangential loads. The testing apparatus developed at the Massachusetts Institute of Technology gives the user the ability to control and measure both the normal and tangential contact loads. This apparatus has been used to conduct fretting fatigue tests on Al 7075-T6 and Ti-6Al-4V. The testing apparatus is described in [25]. A simplified diagram of the test setup is shown in Figure 3-1. These tests provide the necessary information to calculate the elastic stress fields due to contact, which will be discussed in detail in the following sections. The elastic stress fields at the surface due to contact do not completely describe the fretting phenomenon, there are some shortcomings. The stress analysis does not account for the condition of the surface and may not give the correct location of crack initiation.
However, the stress analysis provides a very good starting point for a systematic analysis of the problem.
3.2 Al 7075-T6 Experimental Results
The experimental data provided in the work performed by Birch [11] and Conner [12] is shown in Table 3.1. Table 3.1 provides all the necessary information to calculate the stress fields. The bulk stress, 0-b, for these experiments was applied in the x-direction only, with a stress ratio,
24 /. / /,
P TP
Applied Bulk Load
Figure 3-1: Simplified diagram of the fretting fatigue testing apparatus.
R= -1.
Care must be taken when superposing the bulk stress. As discussed earlier the trailing edge of contact is the point of interest for this analysis; however, during each complete stress cycle the trailing edge becomes the leading edge as the bulk stress is reversed. The trailing edge during the tensile portion of the bulk stress loading cycle becomes the leading edge during the compressive portion of the loading cycle. This location on the specimen undergoes the highest tensile stresses and the largest stress amplitude. The opposite edge of contact endures lower stresses because it becomes the trailing edge of contact during the compressive portion of the bulk loading cycle. It would be expected that the cracks would initiate from the edge of contact which undergoes the highest stresses. This fact is verified in [23] which displays a fretting scar with cracking visible only on one side of the circular contact region. The loading conditions displayed in Table 3.1 were used to calculate the surface stresses at the trailing edge of contact for each test specimen. To better analyze the results of the stress analysis, this work has isolated two series of experiments, which correlate to two different loading conditions. The first series of tests includes those tests where the bulk stress was roughly constant (-83MPa), while the contact loads were varied. Therefore, the effects of the
25 Test P (N) Q (N) Q/(ii x P) O'b (MPa) Nf (Cycles) 1 20.8 15 0.60 83 449,5500 2 30 15 0.42 85 480,000 3 15.6 15 0.80 85 395,000 4 12.5 15 1.0 83 361,000 5 18.5 13.5 0.61 77 551,000 6 16 11.7 0.61 83 530,000 7 8.75 7 0.67 83 3,300,000 (no failure) 8 13 7 0.45 83 10,680,000 (no failure) 9 13.9 10 0.60 83 803,000 10 10.3 7.5 0..61 83 2,940,000 11 7.31 6.6 0.75 83 3,450,000 (no failure) 12 8.33 8 0.80 83 616,000 13 20 15 0.63 69 1,410,000 (no failure) 14 20 21.5 0.89 54 2,186,000 (no failure) 15 20 20 0.83 61 2,500,000 (no failure) 16 24 20.25 0.70 61 2,666,000 (no failure) 17 28 20 0.59 61 2,505,000 (no failure) 18 11.5 8.6 0.62 83 2,608,000 (no failure) 19 20 15 0.63 83 549,000 20 20 15 0.63 70 516,000 21 20 15 0.63 56 1,540,000 (no failure) 22 20 15 0.63 63 2,940,000 (no failure) 23 20 15 0.63 59 1,777,000 (no failure) 24 25 23 0.77 270 297,000 25 25 22.5 0.75 270 135,000 26 22.5 19.5 0.72 230 480,000 27 20 17 0.71 195 575,000 28 20 15 0.63 75 3,050,000 29 20 15 0.63 100 390,000 30 20 15 0.63 125 258,063 31 20 15 0.63 150 181,118 32 20 15 0.63 74 2,497,226 33 20 15 0.63 67 3,200,845 (no failure)
Table 3.1: Al 7075-T6 fretting fatigue experiments
26 Test a (Mm) c/a Uxtot (MPa) oec (MPa) Nf (Cycles) 8 184 0.82 338 248 10,680,000 (no failure) 7 162 0.69 360 264 3,300,000 (no failure) 11 152 0.63 362 264 3,450,000 (no failure) 10 171 0.73 361 266 2,940,000 18 177 0.72 375 280 2,608,000 (no failure) 12 159 0.58 384 285 616,500 9 188 0.74 388 292 803,000 6 197 0.73 405 309 530,000 2 244 0.83 409 317 480,000 1 216 0.74 432 334 449,500 19 213 0.72 435 337 549,000 3 196 0.58 456 351 395,000 4 182 0.0 473 366 361,000
Table 3.2: Al 7075-T6 Constant Bulk load data analysis bulk stress are the same in each test and the difference in life is due only to the difference in contact stress. The second series of tests analyzed are made up of tests with constant contact loading (P = 20 N, Q = 15 N), while the bulk stress is varied, which isolated the effect of the bulk stress on fatigue life.
The results of the stress calculations for the constant Ub series of tests are shown in Table
3.2. Table 3.2 displays the calculated values for the contact radius, a, the ratio of the stick zone radius to the contact zone radius (c/a), the maximum tensile stress at the trailing edge of contact, oxtot and the effective stress due to contact at the trailing edge, oec.
Table 3.2 has been arranged in order from lowest to highest ucc. Note that the fatigue lifetimes as a result are nearly arranged in order from greatest number of cycles to lowest.
This fact indicates that a relationship exists between the surface stresses at the trailing edge of contact and the fretting fatigue life of the specimen. This relationship will be discussed further in the next chapter, where fretting fatigue life prediction methods are discussed. The fatigue life versus effective stress due to contact is plotted in Figure 3-2. Figure 3-2 again demonstrates the existence of a relationship between the fatigue life and the surface stresses due to contact.
Recall that each one of these tests were conducted with approximately the same applied bulk load, therefore the difference in fatigue life is due to the difference in contact loading only. It can be seen that as the contact stresses are reduced, the fatigue life increases. For this bulk loading condition it appears that specimen failure will not occur if a,, is maintained less than
27 380
360 ------o- No failure:
340 ------[L
320 ------
0
------280 2 8------0 ------
260 0 5 105 1 106 1.5 106 2 106 2.5 106 3 10 3.5 106 N (Cycles)
Figure 3-2: Plot of effective stress due to contact versus number of cycles to failure for -b ~ 83 MPa. about 264 MPa, which provides an endurance type stress combination. Also note that in none of the tests was the effective stress due to contact greater than the yield strength of the material; however, in test four, the total stress in the x-direction does slightly exceed the yield strength of the material.
The stress values are not the only important values displayed in Table 3.2; the values calculated for a and c/a are also quite important. They provide analytical values that can be easily checked against the actual scars on the specimens, which is done later in this chapter.
This gives insight into the validity of the tests.
The value of c/a not only provides a means to verify testing procedures, it also is a key parameter in characterizing the nature of the fretting condition. Recall that as c/a goes to zero, global sliding is introduced and when c/a = 1, complete stick is occurring. It can be seen from Table 3.2, that in Test 4 global sliding should be occurring. The c/a value can also provide insight into the existence of a normal pressure threshold discussed by Adibnazari and
Hoeppner [15]. Table 3.3 displays tests with the same value of tangential load (15N), with increasing values of normal load. It can be seen that as the normal load is increased, the maximum tensile stress and the effective stress due to contact actually decrease. Therefore, for
28 Test P (N) Q (N) Uxtot (MPa) -ec (MPa) 4 12.5 15 473 366 3 15.6 15 456 351 14 20 15 435 337 1 20.8 15 432 334 2 30 15 409 317
Table 3.3: Stress variation as the normal load changes for a constant maximum tangential load. a given tangential load, as the normal load is increased, the contact stresses at the trailing edge of contact decrease, and from Table 3.2 and Figure 3-2 the fretting fatigue life should increase.
This provides a rational explanation of why a pressure threshold exists. The stresses decrease as the normal load is increased due to the lowering of c/a. As c/a goes to zero, the relative displacements also go to zero and complete stick results.
The results for the second series of tests, in which the contact loads were maintained constant while the bulk load was varied, are shown in Table 3.4. The data in Table 3.4 is presented in order of ascending bulk load. The values for contact radius, a and c/a are unchanged for
each test, since the contact loads are identical for each test. The values of a and c/a were
found to be 213 tim and 0.72 respectively. The value of c/a was <1, therefore all the tests
were conducted under stick/slip conditions. The most interesting aspect of this set of tests
is the appearance of a fretting fatigue endurance type limit for the given contact conditions. It appears that when the bulk stress is reduced below approximately 70 MPa for the given
contact loading conditions, specimen failure will not result. Figure 3-3 is a plot of the bulk
stress versus the number of failures to cycle (S-N curve) for these contact conditions. This
is not a curve with universal applications. It only applies to the contact conditions used in
this set of tests. A different set of contact load conditions will result in a different endurance
strength. However, combining the data from the first set of Al 7075-T6 tests with this set
of data and the known endurance strength of the material a curve that does have universal
applications can be proposed. This curve is shown in Figure 3-4. In this figure the so called
knock-down factor, which is the factor reduction of the endurance strength of the material due
to fretting, is plotted versus the effective stress due to contact at the trailing edge.
The curve displayed in Figure 3-4 is a very rough approximation of the reduction in the
29 Test a (pm) c/a oxtot (MPa) oec (MPa) Nf (Cycles) Ob(MPa) 21 213 0.72 408 337 1,540,000 (no failure) 56 23 213 0.72 411 337 1,777,000 (no failure) 59 22 213 0.72 415 337 2,940,000 (no failure) 63 33 213 0.72 419 337 3,200,845 (no failure) 67 13 213 0.72 421 337 1,410,000 (no failure) 69 20 213 0.72 422 337 516,000 70 32 213 0.72 426 337 2,497,226 74 28 213 0.72 427 337 3,050,000 75 19 213 0.72 435 337 549,000 83 29 213 0.72 452 337 390,000 100 30 213 0.72 477 337 258,063 125 31 213 0.72 502 337 181,118 150
Table 3.4: Al 7075-T6 stress analysis for constant contact loading
160 -0 ------11------140 ------No failure ------120 -
- - 'U 100 7 -- ---
80 ------7- ---- 60 K-~7------I I0 ------_-LLL 40 + 5 3.5 10 0 5 10 1 10 1.5 106 2 106 2.5 106 3 10 N (Cycles)
Figure 3-3: Number of cycles to failure versus Ub, S - N curve for constant contact load conditions.
30 -b (MPa) oec (MPa) KF 70 337 2.51 83 264 2.12 176 0 1.0
Table 3.5: Data used to plot the knock-down factor curve. endurance strength of the material due to fretting. The knock-down factor is given by:
KF = Jend(no fretting) Uend (fretting)
where rend is the endurance strength of the material. Figure 3-4 was constructed using only three data points; several more series of tests at different bulk loads need to be performed to determine the accuracy of this plot. However, the accuracy of this plot is not the critical issue, the premise on which the plot was constructed is much more important. The plot indicates that KF is not a single number for a given material, but it can be determined if the contact stresses are known. The fact that the strength reduction changes for various contact loads can
easily be seen in Figure 3-2. This idea will be examined closer in the next chapter. The data
points used to make Figure 3-4 are displayed in Table 3.5. When the contact stresses go to
zero, the knock-down factor goes to one.
3.3 Ti-6A1-4V Experiments
The second material analyzed in this work is Ti-6A1-4V [12]. The same test apparatus discussed
earlier was also used in the Ti-6Al-4V fretting fatigue tests, utilizing the same sphere on flat
surface fretting geometry. Table 3.6 provides the necessary material properties to perform the
stress calculations for this material. Fuchs [22] provides fatigue properties for many different
materials and was used along with [11] and [45] to produce Table 3.6.
The loading conditions used to conduct the tests on the titanium alloy are presented in
Table 3.7. The results of the stress analysis are displayed in Table 3.8. The value used for the
coefficient of friction was /y = 0.5 for the displayed results. The fretting pad radius, R, used in
these tests was 0.5 inches versus one inch used for the Al 7075-T6 tests.
A plot of the fatigue life versus the effective stress due to contact is shown in Figure 3-5.
31 3
------2.5
2 U------
1.5
0.5 -50 0 50 100 150 200 250 300 350 a (MPa) ec
Figure 3-4: Plot of the knock-down factor versus effective stress due to contact for Al 7075-T6.
Material Property Value Young's Modulus (E) 115.7 GPa Poisson ratio () 0.321 Coefficient of kinetic friction 0.4-0.5 Monotonic yield stress (oy) 925 MPa Endurance strength (Jend) 525 MPa Fatigue strength coefficient (o-') 1933 MPa Fatigue stength exponent (b) -0.10
Table 3.6: Ti-6A1-4V material properties.
Test P (N) Q (N) Q/([L x P) 0 -b (MPa) Nf (Cycles) 1 50 31.5 1.26 225 691,612 2 50 23.5 0.94 400 109,574 3 50 14.5 0.58 394 107,879 4 50 13 0.52 357 31,540 (interrupted) 5 50 12.25 0.48 298 304,456 6 50 14.5 0.58 300 474,656 7 50 23 0.96 327 143,683 8 30 18 1.20 300 258,576 9 30 9 0.60 300 3,257,000 (no failure)
Table 3.7: Test parameters for Ti-6A1-4V fretting fatigue tests
32 Test a (pIm) c/a oxtt (MPa) o, (MPa) 1 195 0 973 714 2 195 0.39 988 567 3 195 0.75 839 448 4 195 0.78 775 426 5 195 0.80 702 414 6 195 0.75 745 448 7 195 0.43 907 564 8 164 0 904 577 9 164 0.74 682 384
Table 3.8: Ti-6Al-4V stress analysis
This figure includes tests 5-9, where the bulk stress was ~ 300 MPa. Again, the fatigue life increases as the contact stresses are reduced. An endurance type limit is also seen for this material. For Ob ~ 300 MPa, it appears specimen failure will not occur if -ec is maintained less than about 390 MPa. More experimental tests are needed to confirm this observation.
Not enough experimental data was available to plot a graph of the fretting fatigue life versus applied bulk stress.
3.4 Experimental Observations
Observations of the fretting pads and specimens have been made using a scanning laser micro- scope and an optical microscope. The pictures provided in this section were produced using images provided by the scanning laser microscope. The scanning laser microscope made it very simple and quick to perform accurate measurements of the fretting parameters, such as the contact radius, a and the radius of the stick zone, c.
The expressions for the contact radius, a, and the stick zone radius, c, as derived by Hertz
[17] and Mindlin [19] respectively are given in Chapter 2 of this work. The analytical values for these parameters have been calculated for each test conducted on Al 7075-T6 and Ti-6A-
4V and will be compared with the actual specimen values when possible. The results of the comparison are given in the next section.
A second objective of this observational work was to determine the location of crack initia-
tion. To achieve this objective the spherical fretting pads and the unbroken fatigue specimens
33 600
- --- - 55 0 ------
>- No failure
4 0 0 ------
350 6 10 5 10 10 N (Cycles)
Figure 3-5: Plot of effective stress due to contact versus number of cycles to failure for Ti-6Al-4V (Ob -_ 300 MPa). have been analyzed. The fretting pads provide an opportunity to see the influence of the con- tact stresses alone on crack initiation, due to the fact that they are not subjected to an applied bulk stress.
3.4.1 Al 7075-T6
Not all of the specimens from the 33 Al 7075-T6 tests shown in Table 3.1 were available to be analyzed, only about one-half of the tests had pads or specimens that could be analyzed. The fretting pads provided nearly all of the observations because when the specimens failed they usually failed at the fretting scar. It was very difficult to determine the fretting parameters a, and c/a for many of the pads and in some cases it could not be done. The stick and slip zones could not be distinguished for many of the pads that correspond to specimen failure. Table
3.9 shows the comparison between the predicted (Pred. in the table) and actual (Act. in the table) values for the fretting geometry, for those specimens where the comparison was possible.
L and R correspond to the left and right fretting pads respectively. Test 33, which did not produce specimen failure also displays the values for the specimen and are denoted by "(spec)"
34 100 pm
Figure 3-6: Picture of the left fretting pad used for test 33 with the approximate size of the stick zone shown (Al 7075-T6). in the table. It can be seen from this table that the actual contact radius is in general larger than that predicted by Equation 2.1. In some cases the difference is quite substantial. The actual c/a ratios are fairly close to the predicted values, indicating that the value used for P is
accurate. As discussed earlier, a scanning laser microscope was used to observe these specimens and
to produce images of the scars. Figure 3-6 shows the fretting scar on the left pad used for test
33. The stick and slip regions are clearly evident and the eccentricity caused by the applied
bulk can also be seen. The corresponding fretting fatigue specimen is shown in Figure 3-7. The
fretting scar on the specimen is not as visible as the scar on the pad. The fretting geometry is
not as easy to distinguish for specimens that were cycled to failure. It is difficult to determine even the contact radius of the pads for the failed specimens and nearly impossible to determine
the stick zone radius. The other objective of this section was to determine the location of cracking. No cracking
was evident in Figures 3-6 and 3-7. It would have been extremely beneficial if cracking could
have been observed in Figure 3-6, because the stick and slip zones are readily identifiable. As
35 Test a (pm, Pred.) a (pim, Act.) I c/a (Pred.) I c/a (Act.) Cracking 5L 207 200 0.73 * no 5R 207 186 0.73 * yes 12R 159 247 0.58 * yes 15L 213 325 0.55 * no 15R 213 305 0.55 * no 17L 238 248 0.74 * no 17R 238 264 0.74 * no 18L 177 210 0.72 * no 19L 213 264 0.72 * no 19R 213 249 0.72 * no 21L 213 226 0.72 * no 21R 213 202 0.72 * no 22L 213 240 0.72 0.64 no 22R 213 241 0.72 0.65 no 23L 213 230 0.72 0.51 no 23R 213 225 0.72 0.67 no 28L 213 362 0.72 0.79 no 28R 213 * 0.72 * no 29 L 213 * 0.72 * indefinite 29 R 213 * 0.72 * indefinite 30 L 213 * 0.72 * yes 30 R 213 * 0.72 * yes 31 L 213 241 0.72 0.65 yes 31 R 213 265 0.72 * yes 32 L 213 239 0.72 * yes 32 R 213 279 0.72 * yes 33 L 213 238 0.72 0.61 no 33 R 213 257 0.72 0.61 no 33 L (spec) 213 253 0.72 0.69 no 33 R (spec) 213 244 0.72 0.78 no
Table 3.9: Comparison of the analytical versus actual values for the fretting geometry, Al 7075-T6 (* indicates value could not be determined).
36 100 pm
size Figure 3-7: Picture of the left side of the specimen used for test 33 with the approximate of the stick zone shown (Al 7075-T6).
on the Table 3.9 shows, cracking was found on eight of the fretting pads. The cracking found of the left fretting pad for test 32 is shown in Figure 3-8. The cracking is on the right side and slip picture and appears to have started approximately at the edge of contact. The stick zones cannot be distinguished, so the exact location of crack initiation cannot be determined for test from this figure. It is interesting that cracking was found on the fretting pads used test 33. The 32, but no evidence of cracking can be found on the pads or specimen used for was two tests had exactly the same contact loading and the difference in the applied bulk stress in both tests only 7 MPa. It would seem logical that fretting would have initiated cracking this was and an arrested crack would be produced in test 33 due to the lower bulk stress, but
not found.
3.4.2 Ti-6A1-4V
in Table The results of the observations performed on the Ti-6Al-4V test specimens are shown which 3.10. Again it was very difficult to observe the fretting geometry for the specimens the were cycled to failure; however, test 5 provided an interesting specimen. Even though
37 100 pm
Figure 3-8: Picture of cracking on the left fretting pad used for test 32 with the approximate circular area of contact shown (Al 7075-T6). specimen was cycled to failure, the fracture surface went above the scar on the left side of the specimen and preserved the scar. The ratio of c/a was calculated using a coefficient of friction, and calculated values p = 0.5, which produced fairly good agreement between the analytical of c/a. The predicted values for the radius of contact were also in reasonably good agreement with the actual values observed for the tests.
The scanning laser microscope was again used to produce photos of the pads and specimens.
The left fretting pad used in test 5 is shown in Figure 3-9. The slip and stick zones are clearly
visible in this figure, which is unusual for a test that went to specimen failure. The left side
of the specimen used in test 5 is shown in Figure 3-10. This is the specimen discussed earlier
where the fracture plane was above the fretting scar on the left side of the specimen.
It is interesting that cracking cannot be observed in Figure 3-10. It seems reasonable
that cracking would initiate on both sides of the specimen since both sides are under the same the loading conditions. Cracking was observed in the fretting pads of test 2. The cracking in
right fretting pad used in test 2 is shown in Figure 3-11. Cracking can be seen on the left edge of the scar. Cracking was also found on the opposite side of the scar, but it cannot be seen in
38 Test a (ptm, Pred.) a ([Lm, Act.) c/a (Pred) c/a (Act.) Cracking 1 L 195 200 0 * no IR 195 241 0 0.34 no 2 L 195 216 0.39 0.47 yes 2 R 195 247 0.39 * yes 3 L 195 280 0.75 * no 3 R 195 * 0.75 *no 5 L 195 195 0.78 0.76 no 5 R 195 230 0.78 * no 5 L (spec) 195 196 0.78 0.77 no 5 R (spec) 195 175 0.78 * no 6 L 195 252 0.75 * no 6 R 195 243 0.75 * no 7 L 195 252 0.43 *no 7 R 195 * 0.43 * no 8 L 164 189 0 * no 8 R 164 302 0 * no 9 L 164 233 0.74 0.64 no 9 R 164 197 0.74 0.65 no
Table 3.10: Comparison of the analytical versus actual fretting geometry (Ti-6A1-4V).
,wOIIIIV -
.K
100 ptm
Figure 3-9: Picture of the left fretting pad used for test 5 (Ti-6A1-4V).
39 't*c ofl-'
100 Jim
Figure 3-10: Picture of the left side of the specimen used for test 5 (Ti-6A1-4V). this photo. The crack appears to have begun near the edge of contact; however, the slip and stick zones are not distinguishable and the exact location of crack initiation is not certain.
40 100 ptm
Figure 3-11: Picture of cracking in the right fretting pad used for test 2 (Ti-6AI-4V).
41 Chapter 4
Fretting Fatigue Life Prediction Analysis
4.1 Introduction
There are two basic approaches used to predict component life for plain fatigue: the damage
tolerant approach and the total life approach. The damage tolerant approach assumes an initial
flaw exists in the material and the life of the structure is the number of loading cycles it takes
to propagate an initial flaw to a critical length where failure occurs. In this method only crack
propagation is considered. The total life approach considers the whole failure crack formation
process, from initiation to component failure. This work uses both approaches to analyze the
fretting fatigue phenomenon to provide as much insight into the problem as possible.
Developing a life prediction method to be used in fretting fatigue applications is difficult
because the effects of fretting are most damaging during the stages of fatigue crack growth
which have the least developed methodologies: the crack initiation stage and the short crack
propagation stage. Methods do not even exist for plain fatigue to determine the number of
cycles required to initiate a crack and propagate it the distance required to use long crack
propagation techniques. The varying stress fields introduced by fretting make this task even
more difficult.
The deleterious effects of fretting on fatigue were shown in the preceding chapter. The applied bulk stress needed to cause fatigue failure in materials under fretting conditions can
42 300
2 5 0 ------Plain fatigue
2------
150 ------Fretting fatigue
100 ------0
10 5 10 10 N (Cycles)
Figure 4-1: Comparison of the S-N curves for plain fatigue and fretting fatigue for Al 7075-T6 (U-ec = 337 MPa).
be much lower than the endurance strength of the material. The endurance strength of Al
7075-T6 is 176 MPa; however, under fretting conditions failures were observed at bulk stresses
as low as 74 MPa. The damaging effects of fretting on fatigue life is shown graphically in
Figure 4-1 for Al 7075-T6. The constant contact loading series of tests (P = 20 N, Q =15 N) has been used for the fretting fatigue S-N curve shown in the figure. Therefore, the fretting
S-N curve represents only one contact loading condition. The curve could change significantly for a different set of contact load parameters. This will be discussed in more detail later in
this chapter.
The fretting S-N curve has been shifted down considerably relative to the plain fatigue S-N
curve in Figure 4-1. Therefore, cracks are initiating and growing to failure in fretting conditions
where there was no failure for the plain fatigue conditions, even at higher alternating stresses,
which demonstrates the critical role of fretting in crack initiation. The fretting fatigue curve
also exhibits an endurance limit. The existence of an "endurance" type limit for fretting fatigue is discussed in more detail in the following section.
43 4.2 Total Life Approach
The total life approach is further subdivided into two categories; the stress-life method and
the strain-life method. The stress-life method is used in high cycle fatigue (HCF) applications
where the applied stresses are maintained in the elastic range. The strain-life method is used
for low cycle fatigue (LCF) applications where plastic strains make up a significant portion of the total strains in the material. The fretting fatigue phenomenon is a HCF problem and the
stresses calculated in the previous chapter were found to be within the elastic region, so the
stress-life method has been chosen for further investigation. The stress-life approach is still
widely used in industry today.
The stress-life approach is based upon the work of Wihler performed in the 1850's and
1860's, who developed the S - N diagram. The S - N diagram is a plot of the alternating stress, S, versus the number of cycles to failure, N. The work of Wohler clearly showed
a relationship between the applied stress amplitude and the number of cycles to failure and
indicated an existence of an endurance limit for some materials. The endurance limit is the
applied alternating stress below which fatigue failure will not occur. Some materials do not
exhibit an endurance limit. For these materials the endurance strength has been defined as
the stress amplitude which produces specimen failure at 5 x 106 cycles, or more often shown as
2N = 1 x 10 7 reversals. A mathematical expression for the stress-life relationship found by Wohler was not produced
until 1910 by Basquin [26]. Basquin noted that a linear relation existed between the applied
stress amplitude and the number of reversals to failure when plotted on a log-log scale. The
relationship is given by: Ao- = Ora = 07f (2Nf)b (4.1)
where 0a is the applied stress amplitude, a' is the fatigue strength coefficient, b is the fatigue
strength exponent and 2N 1 is the number of reversals to failure. The fatigue strength coeffi-
cient, c'I, is approximately equal to the true fracture strength of the material, Of. The fatigue
strength exponent, b, is generally between -0.05 to -0.12. This equation has been used quite
extensively for many years. The values for a'i and b are known for many different materials and can be found in many text books, including references [24] and [22].
44 The Basquin relationship given in Equation 4.1 can be rearranged to give
Nf = 0.5 . (4.2) 01/b
A modification to the equation above is proposed for fretted components such that the number of cycles to failure is now given by:
Nf = 0.5 ( E, (4.3) 0 1/ where F is the fretting correction factor, which is used to account for the premature initiation of cracking in fretted components. To be consistent with earlier notation, ab has been substituted in for Ua. Equation 4.3 is very simple and if its validity can be proved, it would be a valuable design tool. The critical parameter in the modified Basquin expression is the fretting correction factor, F. The values for the fatigue strength exponent, b and the fatigue strength coefficient, 0-' are taken to be the same as for plain fatigue. They have not been modified to account for fretting. Equation 4.3 can be rearranged such that F is given by:
(2N )' a' F ( 7b . (4.4)
The experimental results given in Table 3.1 have been used to determine F for each test. The analysis of F is separated into two categories; the constant applied bulk stress results and the constant contact stress results. This was done to determine the effect of the contact and bulk stresses on F. F is assumed to be a function only of the bulk and contact stresses. The constant bulk stress results are given in Table 4.1. The values of F for the unfailed specimens were determined using the number of cycles where the tests were stopped; therefore, these values of F are maximum values for the given contact conditions.
Figure 4-2 is a graphical representation of the information displayed in Table 4.1. This figure shows the general trend of the data, as the effective stress due to contact is increased, the fretting correction factor, F, also increases. The plot includes the unfailed specimens in order to display the behavior of F at low contact stresses. The F values for the unfailed
45 Test Nf (Cycles) Yb(MPa) oec(MPa) F 1 449,500 83 335 2.820 4 361,000 83 366 2.896 6 530,000 83 309 2.758 7 3,300,000 (No failure) 83 264 2.189 8 10,680,000 (No failure) 83 248 1.860 9 803,000 83 292 2.617 10 2,940,000 83 266 2.223 11 3,450,000 (No failure) 83 264 2.160 12 616,500 83 285 2.706 18 2,608,000 (No failure) 83 280 2.250 19 549,000 83 337 2.746
Table 4.1: The necessary data to determine F and its relationship with the contact stresses.
Test Nf (Cycles) Ub (MPa) F 19 549,000 83 2.746 28 3,050,000 75 2.449 29 390,000 100 2.379 30 258,063 125 2.001 31 181,118 150 1.748 32 2,497,226 74 2.545 1 449,500 83 2.820
Table 4.2: Fretting correction factors for different bulk stresses, constant contact loads. specimens are only approximations, however they do represent the maximum value that the fretting correction factor could be for these tests. As can be seen in Figure 4-2, the initial slope of the F curve is fairly flat then rises sharply before trailing off again at the high contact stress values. The initial slope may even be less, because the F values for the unfailed specimens are their maximum values. These values may actually be much lower. It must be stressed that these results are for one value of Ub, a different bulk loading will yield a different curve.
The values of F for the constant contact loading tests are given in Table 4-3. Only the tests that were cycled to failure were included in this table. Table 4.2 has also included test number one. The contact loading for this test is nearly identical to the others, P = 20.8 N versus 20.0 N used in the other tests. The plot of F versus O-b is displayed in Figure 4-3 .
From Figure 4-3, it can be seen that for a given contact stress, F decreases as the bulk loading is increased except for the two low bulk load cases (tests 28 &32). These two cases will be discussed again later.
46 3
No Failure: U.
L------0 2.5
------I 0 2
------0 1.5 ------
- 1 ------
0.5 -0.2 0 0.2 0.4 0.6 0.8 Cr (y ec y
Figure 4-2: Plot of the Fretting Correction Factor, F versus the effective stress due to contact, Orb = 83 MPa-
3
UL ------L: 2.8 0
------2.6 ------U. C 0 ------2.4 ------
r ------0 2.2 ------U ------2 ------11
------1.8
1.6 60 80 100 120 140 160 a b (M Pa)
Figure 4-3: Plot of F versus applied bulk stress, while maintaing constant contact loads for Al 7075-T6 (9ec = 337 MPa).
47 180
1. . ...- --.-- .I------
- Failure:------:L- - - - - 1160 4 0 ------
1 0 -0 ------No failure-
8 ---0 ~ ------
60 -50 0 50 100 150 200 250 300 350 a (MPa)
Figure 4-4: Plot of the combination of bulk and contact loading above which fretting fatigue failure will occur for Al 7075-T6.
The constant contact stress analysis and the constant bulk stress analysis will now be combined to provide an estimate of the general behavior of F for various contact and bulk loadings. The knock-down factor idea discussed earlier can aid in this effort. Two thresholds have been found for the two series of tests. Given yb = 83 MPa, specimen failure will not occur if oe < 264 MPa, and given oe = 337 MPa, specimen failure will not occur if c'b < 70 MPa.
These two thresholds can be combined with the endurance strength of the material to produce the failure envelope curve shown in Figure 4-4. Specimen failure is predicted for combinations of contact and bulk loadings which fall above the curve. Figure 4-4 can now be used to analyze Figure 4-2. For ac < 264 MPa fretting will not cause specimen failure if b < 83 MPa; therefore F = 1.0 for these loading conditions. Figure
4-2 can now be redrawn as seen in Figure 4-5.
The behavior of the curve in Figure 4-5 is very interesting. Fretting has no effect on the fatigue life of the specimen for a large range of cc. However, once the threshold value of ac is crossed, the impact on the fretting correction factor, F, is quite significant. Below the threshold, the contact stresses are not high enough to initiate and grow the crack to the length required for long crack propagation by the bulk stress. Once the threshold contact stress is
48 -=83MPa 3 ------2------
0 2. ------E7 ------MEl------LL - - U- 2.5 - --- C 0
0b
Fiur 4 R7s p ff l
The sa aOw st ec
Figure 4-5: Revised plot of F versus foras Al 7075FT6, ireb 83 MPa. reached cracks will grow enough to be controlled by the bulk stress, but the rate at which this occurs is not greatly effected by how far above the threshold the contact stresses are. The steep portion of the F curve appears to be a very narrow region surrounding the threshold.
The slope of the upper portion of the curve is relatively flat. The constant contact loading data in Table 4.2 and Figure 4-4 can now be used to estimate the behavior of F for other bulk loading conditions as shown in Figure 4-6. The threshold values for Ub =100 MPa and 0 b = 125 MPa were determined using the failure envelope curve.
The shape of the curves were assumed to be similar to that found for O'b = 83 MPa. It can be seen that the F curves for the 100 and 125 MPa bulk loading cases pass through the previously determined values of 2.38 and 2.0 at a,,c 337 MPa. Tests 28 & 32 discussed earlier provide results that support the validity of the curves pre- sented above. The F value found for these two cases fell below the curve for the results plotted in Figure 4-3. The threshold cc determined from Figure 4-4 is 311 MPa for ab = 74 MPa.
Therefore, the contact stress of 337 MPa is approaching the threshold value and could be on the steep portion of the F curve, which would result in an F value lower than the expected
49 - 3 3
U- 2.5 -
1 . --
100 150 200 250 300 350 400 a (M Pa) ec
Figure 4-6: Plot of the predicted behavior of F versus a-, for various bulk loadings (Al 7075-T6). peak value.
For a given set of loading conditions Figure 4-6 can be used to determine F, which can then be substituted back into equation 4.3 to determine the component life. However, Figure 4-6 is only a rough approximation of the behavior of F for various loading conditions. Many more tests need to be conducted to determine a better estimate for the behavior of F. The threshold contact stress values are the most critical values because it is there where the correction factor is changing significantly and hence where fretting fatigue life is changing the most. The advantages of this method are that it is simple in form and utilizes the familiar stress- life approach widely used in industry today. However, to implement this method will require considerable experimental work for many different materials. The method also does not provide information on the location of crack initiation. The method uses the contact stresses at the edge of contact, but this is not always the location of crack initiation.
The analysis performed in this section has produced some very interesting results which are:
9 A rough approximation for the component failure envelope as a function of the bulk and
contact stresses has been determined and is shown in Figure 4-4.
50 1.8 E5 U------0 1.6
C 0 1 .4 CD 0 1 .2 ------0 IM E) 1 ------U-
0.8 - 100 0 100 200 300 400 500 600 a (M Pa) ec
Figure 4-7: Plot of F versus -ec for Ti-6A1-4V (ub ~ 300 MPa).
" The slope of the top of the F curve is relatively flat indicating that the fatigue life is not
greatly affected by changes in the effective stress due to contact in this region.
" F acts almost as a step function making the determination of the threshold stress values of great importance.
The same analysis was also performed for the Ti-6A1-4V tests recorded in the previous chapter. Tests 5-9 from Table 3.7 were used to determine F as a function of -ec for ab ~ 300
MPa. The results are shown in Figure 4-7. This curve is based on limited experimental data, but the same type of behavior as seen with the aluminum alloy seems to be present for the titanium alloy. For the given bulk loading, F = 1 until a-ec ~ 390 MPa then it quickly jumps to ~ 1.7.
4.3 Damage Tolerant Approach
4.3.1 Al 7075-T6
The second life prediction approach used to analyze the fretting fatigue problem is the damage tolerant approach. Using the experimental results from the preceding chapter, this work
51 attempts to determine the relative magnitudes of the crack nucleation lifetime and the long crack propagation lifetime. The crack nucleation lifetime used in this work is the life spent initiating and growing the fatigue crack to the length where the fretting stresses are approximately zero.
This approach is used by Szolwinski and Farris [30] in their fretting fatigue life prediction method. The long crack propagation life then is only influenced by the bulk stress. Linear
elastic fracture mechanics can be used to determine this life. This long crack propagation
life can then be compared with the total fatigue life from the experiments to determine the
relative magnitudes of the two stages of fatigue life. The approach is demonstrated using the
Al 7075-T6 experiments and is applied in the following section to the Ti-6Al-4V experiments.
Paris, Gomez and Anderson [31] in 1961 and Paris and Erdogan[32] in 1963 demonstrated that a power law relationship exists between the crack growth increment da/dN and the stress intensity factor range AK. This relationship is given by:
da = C(AK)' (4.5) dN
where C and m are scaling constants, and
AK = Kmax - Kmin, for Kmin > 0
AK = Kmax, for Kmin < 0.
Observations were made of the failed test specimens to determine the shape of the propagating
crack. It was determined that the cracks could be best characterized as surface elliptical cracks
with a ratio of a/c = 0.9. A diagram of this geometry is shown in Figure 4-8, with the thickness
and width of the fatigue specimens also given.
The stress intensity factor equation for the surface elliptical crack geometry was determined
by Newman and Raju [33] in 1981. For a finite plate subjected to a remote uniform tensile
stress, S, the stress intensity factor is given by:
Kr a ,a, , (4.6)
52 t =5mm
-C 2c
2V=10mm
Figure 4-8: Diagram showing the elliptical crack geometry. where Q=1±1.46 .(5 a Q=I+1.464 -) 1 for-<1. c C
The expression for F was obtained using a systematic curve-fitting procedure [33]. Note this
F is not the same F described in the previous section and a in these expressions refers to the
crack geometry and not the contact radius. The expression for F is given by:
2 F = [M1 + M 2 (a) + M 3 (a)4] fpgf. (4-7)
where M1 = 1.13 - 0.09 (a) (4.8)
0.89 M2 = -0.54 + 0. (4.9) 0.2 + (a/c) 1.0 a2 M = 0.5 - +14 (1.0 - (4.10) 3 0.65 + (a/c) C
g = 1+ 0.1 +0.35 (a)2] (1 - sin 0)2 (4.11)
53 Parameter Value m 4 C 8.6 x10- 12 ao 850 pum af 2.5 mm t 2.5 mm
Table 4.3: Values used in the Paris life calculations for Al 7075-T6. Consistent units for C are meters and MPa.
[a2 1/4 fo = [(a)cos2 0+ sin 2 #/ (4.12)
fw = [sec ( 7'7 1/2. (4.13)
The function, f, is the finite width correction. The above equations can now be used to find K given any a. The expression for the stress
intensity factor is now substituted back into the Paris form, Equation 4.5 and the equation is numerically integrated from an initial crack length ao to the critical crack length af to find the
crack propagation life. Using # = 90', the propagation life, Nfp is given by:
1 fa da NfP = C A-472J 244 1 ,a(4.14) 4 4 ]4 ~C(Z ) 2 Ja0 (0.67) a2 1.05 +0.27 (a)2 (0.145(4 [sec (11a
The values used in the propagation life calculations are displayed in Table 4.3 [35]. The
value used for ao is approximately four times the contact radius for the tests conducted on the
aluminum alloy. The stresses due to contact decay away very quickly and are approximately
zero at a depth of 1.5a. However, if 1.5a is used for ao, the calculated propagation life is greater
than the experimental life, which cannot be true. Therefore, ao was set equal to 4a. The
value used for ao must also be long enough, such that it includes at least a few grain diameters.
The average grain size for the Al 7075-T6 specimens was 60 pm [23], so the ao used in these
calculations encompasses several grain diameters. As can be seen from Table 4.3, only one-half
the thickness of the specimen was used in the calculations. This is due to the presence of two
cracks in the specimens. When the fracture surfaces of the failed specimens were analyzed, two cracks were observed. One on each side of the fatigue specimen.
Using Equation 4.14, the long crack propagation life was calculated for each test, using Ob
54 for Ao-. The stress intensity factor range was also calculated for each test using Equation 4.6, at a = ao, and S = Ub to examine if AK > AKO, the threshold stress intensity factor range at this crack length. The AKO for Al 7075-T6 at R = -1 appears to be approximately 2-3
MPav/ii. This value was determined from a plot in [34]. The plot did not contain R = -1 crack growth data; however, using the displayed trends for the other R ratios an estimate for the AKO for R = -1 was made. The calculated values of the propagation life, Nf, and the stress intensity factor range at a = ao for each test are given in Table 4.4.
As can be seen from Table 4.4, the long crack propagation life is a very significant portion
of the fretting fatigue life for many of the failed specimens. For these specimens, a very small
portion of the life was used to nucleate cracks. The calculated AK values in Table 4.4 provide insight into some of the tests where component failure did not occur. It can be seen that for Tests 14-17 and Tests 21-24, AK < 2.5 MPay'iii for each of these tests, which is very close to
the threshold value for this material.
4.3.2 Ti-6A1-4V
The method used for Al 7075-T6 above is applied to Ti-6A1-4V in this section. The crack
geometry for the titanium alloy specimens was very similar to that found above. Again elliptical surface cracks were found; however, cracking was found on only one side of the specimen for this
material. The values used to calculate the propagation life for this material are shown in Table
4.5 (a/c corresponds to the elliptic crack geometry). The values used for ao are approximately
1.5a and 4a. The first value of ao ~ 1.5a, corresponds to the depth at which the tensile stresses
due to contact are approximately zero. The second value of ao - 4a is used to be consistent
with the calculations performed earlier for Al 7075-T6.
The values for m and C were determined using curves found in [37] and [36]. The fatigue
crack growth curve in [36] was for solution treated and over-aged (STOA), Bi-modal Ti-6A-4V, while that found in [37] was for Bi-modal Ti-6A1-4V with no heat treatment. Both curves were
made using R = 0.1. The values found for C and m for the two cases are shown in Table 4.6.
The experimental results displayed in Chapter 3 are for Bi-modal Ti-6A1-4V with no special
heat treatment, at R = -1. The value of m for this section was set to 5.0, since the applied stress ratio has a small effect on m. The value of C was determined by approximating the
55 Test I Experimental Nf (Cycles) Nf, (Cycles) I AK (MPaVii) 1 449,000 408,000 3.17 2 480,000 371,000 3.25 3 395,000 371,000 3.25 4 361,000 408,000 3.17 5 551,000 551,000 2.94 6 530,000 408,000 3.17 7 3,300,000* 408,000 3.17 8 10,680,000* 408,000 3.17 9 803,000 408,000 3.17 10 2,940,000 408,000 3.17 11 3,450,000* 408,000 3.17 12 616,500 408,000 3.17 13 1,410,000* 855,000 2.64 14 2,186,000* 2,280,000 2.06 15 2,500,000* 1,400,000 2.33 16 2,666,000* 1,400,000 2.33 17 2,505,000* 1,400,000 2.33 18 2,608,000* 408,000 3.17 19 549,000 408,000 3.17 20 516,000 807,000 2.68 21 1,540,000* 1,970,000 2.14 22 2,940,000* 1,230,000 2.41 23 1,777,000* 1,600,000 2.26 24 297,000 3,650 10.32 25 135,000 3,650 10.32 26 480,000 6,920 8.79 27 575,000 13,400 7.46 28 3,050,000 612,000 2.87 29 390,000 194,000 3.82 30 258,063 79,300 4.78 31 181,118 38,300 5.74 32 2,497,226 646,000 2.83 961,500 2.56 33 I 3,280,000* * No failure
Table 4.4: Long crack propagation life for Al 7075-T6.
56 Parameter Value a/c 1.0 t 5.0 mm 2W 5.0 mm m 5.0 C 7.0x10- 1 4 ao 290 pm, 780 pm af 5 mm
Table 4.5: Values used for the propagation life calculations for Ti-6Al-4V.
STOA, Bi-modal Ti-6A1-4V Bi-modal Ti-6A1-4V C 8.13x10- 4 1.8x10-13 m 5.12 5.0
Table 4.6: Values of C and m for STOA, Bi-modal Ti-6A1-4V and Bi-modal Ti-6Al-4V (units are meters and MPa). effect of R on C.
Using the values displayed in Table 4.5, the propagation life was calculated for the nine tests shown in Table 3.7. The propagation lives corresponding to the two different values of
ao are shown in Table 4.7. As can be seen from the table, the propagation life correlates very well with the actual experimental life of the failed specimens when ao = 1.5a is used. Using
ao = 1.5a, the difference in fatigue life for the various tests can be attributed almost entirely to the difference in propagation life. Using ao = 4a results in much lower values of Nfp. Table
3.7 also displays the calculated AK values at ao = 1.5a. The threshold stress intensity factor range, AKO, is approximately 3-7 MPaViii [37].
Test Nf (Experiment) Nfp(ao = 1.5a) Nfp(ao = 4.0a) AK (MPav/i,ao = 1.5a) 1 691,612 1,010,000 134,000 4.79 2 109,574 56,800 7,540 8.51 3 107,879 61,200 8,130 8.38 4 31,540 (interrupted) 100,000 13,300 7.60 5 304,456 247,000 32,900 6.34 6 474,656 239,000 31,800 6.38 7 143,683 156,000 20,700 6.96 8 258,576 239,000 31,800 6.38 9 3,257,000(no failure) 239,000 31,800 6.38
Table 4.7: Ti-6A1-4V long crack propagation life.
57 Chapter 5
Palliatives
5.1 Introduction
The work performed in the previous chapters provide the necessary background for a discussion of fretting fatigue palliatives. The methods developed earlier provide the criteria that can be used to determine how and why certain palliatives work while others do not.
This work will focus on two fretting fatigue palliatives: shot peening and coatings. Both
have been shown to be beneficial in increasing fretting fatigue life. Chivers and Gordelier [41]
provide a detailed study of several possible palliatives including shot peening and coatings.
5.2 Shot Peening
Shot peening has been used for many years as a way to improve the fatigue properties of
components in plain fatigue. The benefits of shot peening were found quite by accident by
the Buick Motor Division of General Motors Corporation in the late 1920's [421. It was found
that valve springs that had been blasted with grit to remove scale had greatly improved fatigue
properties. It was first thought that the benefits derived from shot peening were due to the work hardening of the surface. However, it was soon discovered that the benefits were caused
by the residual stresses introduced at the surface due to shot peening. In the shot peening process, relatively hard particles are shot against the metal being treated.
The particles are generally spherical in shape and are given enough velocity such that the surface
58 Fretting & Plain Fatigue (shot peened)
Plain Fatigue (not peened)
Fretting Fatigue (not peened)
Cycles to Failure
Figure 5-1: Plot showing approximate relationship between shot peened and unpeened speci- mens for plain and fretting fatigue. of the metal is indented during impact. The indentations occur due to local plastic yielding and as the surface material attempts to expand, it is restrained by surrounding material below the surface which was not deformed plastically. This restraint of the plastically deformed surface layer causes it to be compressively strained and is residually stressed in compression.
There have been numerous studies conducted showing the benefits of shot peening for plain fatigue conditions. Shot peening has also been demonstrated to be effective in fretting fatigue conditions as well, which is discussed in [38], [39] and [41]. In many materials, shot peening has been shown to increase the fatigue strength such that the fretting fatigue strength is approximately the same as the plain fatigue strength. A simple plot displaying the typical relationship between shot peened and unpeened specimens for fretting and plain fatigue is shown in Figure 5-1. This figure is based upon the work of Bignonnet [39], which studied the effects of shot peening on fretting fatigue. However, this figure is not true for all materials. It is only meant to display the benefits typically observed in shot peened materials.
As discussed earlier, the benefits of shot peening are derived from the residual compressive stresses introduced into the material during the process. The maximum value of the residual compressive stress, often called the magnitude, is not affected by the shot peening process as long as the shot used is at least as hard or harder than the material being treated [43]. The magnitude of the residual compressive stress is material dependent, but has a value of at least one-half the yield strength for that material [43]. The depth of the compressive layer is the
59 other important parameter in shot peening. The depth of the compressive layer is a function of the hardness and ductility of the peened material, the state of strain in the material when the peening was performed and the characteristics of the shot and shot stream [42]. A typical residual stress profile due to shot peening and the residual stress profile used in this analysis are shown in Figure 5-2. The residual stress profile used in this analysis assumes a flat maximum compressive stress, which goes directly to zero at a given depth. The assumed residual stress profile should actually have a tensile region to balance the residual compressive stresses to satisfy equilibrium. This is shown in Figure 5-3, where -,c is the residual compressive stress and or is the residual tensile stress required for equilibrium. For equilibrium to be satisfied area Al must equal area A2. Therefore the area balance is given by:
(pd)(O-rc) = (t - pd)(-,t), (5.1) and the ratio of the residual stresses is given by:
rt = pd (5.2) c-rc (t - pd)
If (t - pd) > (pd) then Urt -+ 0. It is assumed in this work that (t - pd) > (pd), therefore the residual tensile stress has been neglected. Note that the actual maximum compressive stress occurs at some depth below the surface. The actual depth of residual compressive stresses for various materials is given in [43] as well as some residual stress profiles.
The benefits derived from shot peening can also become detrimental to the material if the
residual compressive stresses are somehow relieved. The process which caused the residual
stresses to be induced into the material also damages the surface of the material and may even
introduce microcracks. These microcracks could drastically reduce fatigue life if the residual
compressive stresses are relieved. This has been shown to be the case in both plain fatigue and fretting fatigue. Gray, Wagner and Lftjering [44] show the detrimental effects on plain
fatigue life of shot peening when the residual compressive stresses are thermally relieved. The
fatigue strength of the material is significantly reduced for the shot peened, stress relieved
specimens as compared to the original electrolytically polished specimens. Figure 5-4, from
[44], demonstrates this effect. In Figure 5-4, SP is shot peened, EP is electrolytically polished
60 Assumed residual
C/) stress profile
0 ------e-
0 Typical residual 15Z stress prof.ile
Depth
Figure 5-2: Typical stress profile after shot peening and the assumed stress profile used in this work.
Ift IA pd = peening depth
t = material thickness A2 -
P (rec Compressive Stress 0 Grt Tensile Stress
Figure 5-3: Residual stress equilibrium diagram.
61 SP+EP sw 0,4 SP VIM,
SP*SR+EP
CYCLES TO FAILURE
Figure 5-4: Diagram showing the detrimental effects of shot peening if the residual stresses are relieved [44]. and SR is stress relieved. From this figure it can be seen that the fatigue strength for the shot peened and stress relieved specimens is about one-half that of the electrolytically polished specimens at 1x107 cycles, the reduction in fatigue strength is quite significant and certainly cannot be ignored. The shot peened and stress relieved curve shifts down relative to the electrolytically polished specimen curve indicating that the difference between the two curves is in the crack initiation stage, just as was found for fretting. The material used in [44] was
Ti-6A1-4V with a fine lamellar microstructure and the temperature was 20 'C for this set of test results. The residual stresses were relieved thermally using a heat treatment of 600 'C for one hour. These results were for plain fatigue experiments. Bignonnet [39] displays similar results for fretted specimens. The fretting fatigue strength of the shot peened, stress relieved specimens was similar to that of the unpeened specimens. Waterhouse [38] found similar results in his experimental work. These works demonstrate the importance of the compressive residual stresses and the importance of not allowing them to be relieved.
Laser shot peening does not have the shortcomings discussed above for the typical shot peening process. In the laser shot peening process the surface is unchanged by the peening
62 process. Shock waves are introduced into the material that are capable of plastically yielding the material and producing compressive residual stresses.
The residual stresses can be relieved either mechanically or thermally. The approximate temperature at which the residual compressive stresses begin to dissipate for various materials can be found in [43]. The residual stresses are relieved mechanically if the total stresses in the material exceed the dynamic elastic limit of the material. The total compressive stress is made up of the shot peening residual stress and the stress due to external loading. The residual stress is relieved by the amount that the total compressive stress exceeds the dynamic elastic limit of the material.
The objective of this section is find the optimum peening depth for various fretting fa- tigue loading conditions. The magnitude of the residual compressive stresses are fixed by the material, therefore the designer has no control over this aspect of the shot peening process.
However, the depth of residual stress can be influenced by variations in peening parameters as shown in [43]. Therefore, the depth of peening becomes the variable design parameter that can be optimized by the designers.
5.2.1 Material Yielding Analysis
A systematic analysis of the shot peening process applied to the sphere on flat surface contact geometry has been conducted in this work. The material used for this analysis is Ti-6Al-
4V. The pertinent material properties for Ti-6Al-4V are listed in Table 3.6. The first task was to find the conditions when the residual compressive stresses would begin to be relieved.
According to [43], the compressive stresses begin to dissipate thermally at 480 C for Ti-6A-4V.
Mechanical stress relieving begins to occur when the material stresses exceeds the elastic limit
of the material. The stress fields for a sphere fretting against a flat surface have already been
discussed in Chapter 2 of this work. The surface stresses for this geometry for A17075-T6 are
displayed in Figure 2-2. The stress profile for Ti-6A1-4V is almost identical, the magnitudes
are different, but the shape of the curves are the same.
The analysis performed in this chapter assumes the onset of global sliding. This condition
provides the highest stresses possible for a given contact loading. If global sliding is assumed, the normal load, P, becomes the critical contact parameter since the tangential load, Q, will
63 always be equal to pP. The loading which causes the total effective stress, O-etot, to become equal to the yield strength is when material yielding is assumed to occur and is shown in Figure
5-5. The data used to make this figure were determined using a MATLAB script file. A
MATLAB routine was written which first scans the surface of the material, then incrementally scans below the contact surface calculating the stresses at each increment, producing a matrix of stresses. The surface is scanned along the y = 0 line, where maximum material stresses are.
The increment used in the x-direction is 0.02a. The increment in the z-direction is 0.025a.
The program then finds the maximum effective stress from the matrix of stresses determined during the scan. The stresses at the contact edges are calculated separately. The maximum stress is then compared to the yield stress. If the maximum effective stress is less than the yield stress, the contact loading is increased and the stress calculations are performed again.
The process is continued until the contact loading which produces yielding is determined. This was done for five different coefficients of friction and eight different bulk stress conditions. The stresses at the edge of contact are calculated separately because the stresses are indeterminate at the contact origin (x = y = z = 0) and the origin would be included in the surface scan if both contact edges are also included. The program used for this analysis is shown in Appendix A.
In Figure 5-5 the yielding curves have been plotted, which show for a given bulk stress,
Ob and a given coefficient of friction, yt, the minimum contact pressure, Po, that is needed to cause the total effective stress to exceed the yield stress of the material. For example if it = 0.7
2 8 and Ub = O.ly then the maximum contact pressure which produces yielding is ~ 0. ay. This example is shown in Figure 5-5. The maximum contact pressure is normalized by the yield strength of the material in this plot. A residual stress of -0.850y is used for this study. This value for the residual compressive stress was determined using an actual shot peening stress profile from [44] and a graph of maximum compressive residual stress versus tensile strength in [43]. As can be seen in Figure 5-5, the contact loading which causes material yielding increases as the applied bulk stress increases until Ub = 0.4c-y. The leading edge of contact, where the compressive stresses due to contact are very high, is the location of yielding for applied bulk stresses less than one-half the yield strength. The location of yielding shifts to the trailing
64 1.2 b y -. e =o.o b-- 1I -~ ------
b Y 0.8 1a 0.2 b - ga 0.3a
ed,w due A-r,. 0.4 o S 0.6 ------NI ------~ S a =0.5cy >4 0.6a 0.4~~ X ------
0.2 ------
0
0.2 0.4 0.6 0.8 1 1.2
Figure 5-5: The Combination of contact and bulk loading which causes the total effective stress to equal the yield stress plotted versus the coefficient of friction. edge, where the tensile stresses due to contact are very high, when ab ! O.5uy. Therefore, yielding is controlled by the compressive stresses until the applied bulk stress reaches a value of one-half the yield strength. This explains why the maximum contact pressure which causes 4 yielding actually increases with increasing bulk stress when ab < 0. ay. Figure 5-5 also shows the effect of friction. As the coefficient of friction increases, the contact pressure needed to
cause yielding decreases. It can also be seen that the contact loads needed to cause yielding are
relatively low for the coefficient of friction typical for Ti-6A1-4V (0.4-0.5); therefore, residual
stress relieving due to yielding cannot be ignored. Yielding first occurs at the surface for [ > 0.3.
Since yielding usually occurs first at the surface, the depth of yielding must also be found.
5.2.2 Depth of Yielding
The depth of yielding must be found to determine to what depth the residual stresses are being
relieved. The von Mises effective stress is again used to determine the yield conditions, i.e.
when aetot oy the material is considered to have yielded. A MATLAB routine was also
written to perform this task. The routine used to perform this task is similar to the previous
65 1
0.8 ------7 ------
-401 X
0.6 ------X
'U X-
X- Jer -X ------b Y ------0.4 ------jeF e-0- =.o b Y
b Y
------0.2 ------
X 0 0.2 0.4 0.6 0.8 1 1.2 1.4 P / a- 0 Y
Figure 5-6: Plot of yielding depth, z versus maximum contact pressure for various bulk loadings (P = 1.0). one. Again the surface is scanned using an increment of 0.02a in the x-direction with y = 0 for all points. The stresses are then determined below the surface using an increment of 0.02a in the z-direction, which results in a matrix of stresses. The stresses at the edge of the contact are not calculated separately in this routine because the stress profile is much smoother at the edges below the contact surface as shown in Figure 2-3. The depth of maximum stress is found and this depth became the starting point for the iteration to find the depth of yielding. The depth of maximum stress is always the surface for p > 0.3. The scanning process begins again at the depth just determined using a reduced increment in the z-direction of 0.01a. The depth is increased until the effective stress is no longer greater than the yield strength of the material.
Figure 5-6 displays the results of this analysis for A = 1.0. The depth, z is normalized by the radius of contact, a, the maximum contact pressure, P, is normalized by the yield strength of the material, oy. Again the residual stress used in this analysis was -residual = -0. 8 5 ay. The
MATLAB script file used in this section is shown in Appendix B.
The results displayed in Figure 5-6 follow the expected trends, the depth of yielding increases as the contact load is increased and decreases as the bulk loading is increased until a - 0.5oy.
66 0.8 ------
- xe S ------
-0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 P / a 0 Y
Figure 5-7: Depth of yielding versus maximum contact pressure for various bulk stresses (,a = 0.8).
The depth of yielding decreases with increasing bulk stress due to the large residual compressive stress assumed in the analysis. This same analysis was repeated for three more coefficients of friction, ft = 0.8, pt = 0.5 and yt = 0.3. The results for the three other coefficients of friction are shown in Figures 5-7, 5-8 and 5-9 respectively. As can be seen in these figures, the depth of yielding is reduced significantly as the coefficient of friction is lowered, which again points to the possible benefits of coatings in fretting situations. When the coefficient of friction is reduced to 0.3, there is very little yielding even at very high contact loadings for cab > 0.4. The depth of yielding curves developed in this section can be very useful for design, specif- ically in the material treatment stage of design. Designers could use these plots with the known contact and bulk loadings and the coefficient of friction and very quickly determine the minimum depth of shot peening to ensure that the compressive residual stresses are not mechanically relieved. For example, if pt = 0.8, Po/o% = 0.8, and ab = O.lcoy, the depth of yielding, z/a ~ 0.35. Therefore, the minimum depth of shot peening to avoid mechanical relieving of the residual stresses is z/a > 0.35. This example is shown in Figure 5-7. The depth of yielding curves provide important design information, but they only provide part of
67 0.7
0.6
El-- a =-. 0.5 - b Y - ---
0.4 a =AC
bbY Y
0.3 - + + -X -a =.2a b Y
0.2 b Y ------_X ------6a =.4a b Y 0.1 ------a=.5cyb Y
0 - -- a =.6a b Y -0.1 0.2 0.4 0.6 0.8 1 1.2 1.4 P /Cy 0 Y
Figure 5-8: Depth of yielding versus maximum contact pressure for various bulk stress loadings (p = 0.5)
0.6 -- T '---'-'-'-'---
0.5 E a =-.1
0.4 b Y
0.3 -~--- ~-~~~ ~--- ~------~ b Y 0.2 b ------b V - A-a =.4a 0.1 ------X- a b=.5a Y
0 --- a =.3a
-0.1 0.2 0.4 0.6 0.8 I 1.2 1.4 P /Cr 0 Y
Figure 5-9: Depth of yielding versus maximum contact pressure for various bulk stress loadings (u = 0.3)
68 the information needed to determine the optimum shot peening depth. These curves provide the minimum shot peening depth to avoid residual stress relieving but they do not provide information pertaining to the tensile stresses at the crack tip. An additional analysis is needed to provide this information, which is discussed in the next section.
5.2.3 Maximum Principal Stress Analysis
The predicted success of shot peening is based upon two basic assumptions. First, a crack will
not propagate unless a tensile stress forces it open near the tip [43]. The second assumption
is that a crack tip will not open while a compressive force is acting upon it [43]. These
assumptions provide the criteria that is used to determine the optimum shot peening depth.
Using the two assumptions stated above, the depth of peening to prevent crack propagation
should be the depth at which the maximum principal stress, a-1 < 0. Therefore, inside this layer of material affected by shot peening, no region of tensile stresses will exist. This does
not mean that the entire depth of the material will have no tensile stresses. This only applies
in the shot peened region, but that is enough to stop the crack from propagating.
A MATLAB program similar to the two discussed above was written and is shown in
Appendix C. The surface is scanned along the x-direction determining the stresses at each
point using an increment in the x-direction of 0.02a, again y = 0 for all points. At each point
the maximum principal stress, -1 , is determined creating a vector of maximum principal stresses. The maximum of this vector is then found and compared to zero. If the maximum of the vector
is greater than zero, then the depth is increased by 0.01a and the process continued until the
maximum principal is less than or equal to zero. This depth then became the required shot
peening depth. The shot peening depth, z, which produced a maximum principal stress less
than or equal to zero was found for many different loading conditions. Just as in the yielding
depth curves, the depth is plotted versus maximum contact pressure for several different values
of applied bulk stress. Figure 5-10 displays the depth z, at which u1 < 0, for yu = 1.0. Again
the depth has been normalized by the contact radius, a and the maximum contact pressure, Po, is normalized by the yield strength of the material. This analysis was repeated for three
other coefficients of friction, y = 0.8, ft = 0.5 and p = 0.3. The curves for ti = 0.8 and ft = 0.5
are shown in Figures 5-11 and 5-12 respectively. The curves for ft = 0.3 are not displayed
69 0.55
0 .5 ------0---- . ------00- E b
0.45 ------+ 7------+ 4 -+ ------a -. - X c .y
b
0.3 0.2 0.4 0.6 0.8 1 1.2 1.4 P /a 0 Y
Figure 5-10: The depth at which the maximum principle stress is less than or equal to zero plotted versus maximum contact pressure, for magnitudes of bulk stress (p = 1.0). because the required peening depth was less than for 0.l0a for even the highest load conditions analyzed. The plots again show the advantages of lowering the coefficient of friction. There is a significant reduction in z/a as the coefficient of friction is reduced. As expected the required shot peening depth increases as the applied bulk stress, crb is increased. As seen in the figures, 4 the increase in peening depth with increasing crb becomes quite significant when Jb> O. cry. The plots generated in this section cannot be used in isolation, they must be integrated with the plots from the previous two sections of this analysis. Together, these sections provide the designer the information necessary to determine the optimal shot peening depth for the given loading and material conditions.
5.2.4 Integrated Yielding and Maximum Principal Stress Analysis (Opti- mum Shot Peening Depth)
The depths determined from the two preceding sections provide two different threshold val- ues. The depths output from the material yielding analysis provide the minimum depth to which the specimen must be peened to prevent mechanical relieving of the residual compres-
70 0.4
b y 0.35 ------_------
- y ------b 0.3 ------e- . - X-- a =.2ay b V N -- + -a.3a ------+------X- --0A-*-- A b Y 0.25 - - b V -0 -- -- 0 a=.50 0.2 7-- - - - Ea6a
0.15 0.2 0.4 0.6 0.8 1 1.2 1. '4 P / a 0 Y
Figure 5-11: The depth at which the maximum principal stress is less than or equal to zero plotted versus maximum contact pressure for various magnitudes of bulk stress (t = 0.8).
0.2
-a 0.18 ------Ela =-.1a b Y 0.16 -- E a=0 ------b 0.14
N -- 0.12 A - - -- a =.3a b Y 6---a=.3a b y 0.1 c-- =.5a + + -- - -K b y
------At0 a =.5a 0.08 b y
0.06 0.2 0.4 0.6 0.8 1 1.2 1.4 P / a 0 Y
Figure 5-12: The depth at which the maximum principal stress is less than or equal to zero plotted versus the maximum contact pressure for various magnitudes of bulk stress (p = 0.5).
71 sive stresses. The depth of yielding analysis determines the minimum peening depth to ensure that the residual compressive stresses remain in the material. The maximum principal stress analysis assumes the residual compressive stresses are still present and determines the peening depth required to prevent crack growth. Therefore, the depth of yielding analysis provides a minimum shot peening depth, but does not provide all information necessary to determine the optimal peening depth. The remaining information is provided by the maximum principal stress analysis, which determines the peening depths at which 0u < 0. This analysis assumes the crack will not grow in a region of pure compressive stresses. The two analysis must be combined to form the complete picture, which has been done in Figure 5-13 for P = 1.0 and
Ub = 0.4cry. As can be seen from Figure 5-13 the limiting curve depends upon the contact loading, a cross-over point occurs at (Po/ay) ~ 1.15. If (Po/-y) < 1.15 the maximum prin-
cipal stress analysis is the limiting case and controls the optimum peening depth, however if
(Po/o-y) > 1.15 the depth of yielding analysis becomes limiting and determines the required peening depth. This example shows the importance of examining both criteria. An example on how to use the figure is also displayed in Figure 5-13. If (Po/-y) = 0.8 then the optimum
peening depth, z/a ~ 0.4. The designer may choose to shot peen to a slightly greater depth to ensure no tensile stresses exist in the material at that depth.
A second example is given in Figure 5-14 for 4 = 0.8 and b = .ly. As can be seen in
this figure, the intersection between the two curves occurs at a much lower contact pressure
than in the previous example. Contrary to the previous example, the depth of yielding is the
limiting curve over most of the contact loading space for the given ft and O-b. These curves can
be generated quite easily from the figures provided in the previous two sections for many other
combinations of pt and Ub, however only two examples will be displayed in this work. In Figure
5-14 the optimum peening depth, z/a, was again found for a contact load of (Po/-y) = 0.8. In this example the optimum peening depth was found to be; z/a ~ 0.36. The shot peening
depth found in this example would be the minimum recommended because material yielding is still occurring at this depth. The material is just at the yield point, but it is recommended
that the material be shot peened to a slightly greater depth to ensure no material yielding and
hence, no mechanical residual stress relieving will occur.
72 0 .6 ' ' ' ' ' ' ' ' -
0 .5 ------
0.4 -
------0 .2 ------
E3 max analysis ------1------. ------0.1 - -e yielding analysis
0 - -- 0.2 0.4 0.6 0.8 1 1.2 1.4 P / a 0 Y
Figure 5-13: Optimum peening depth plotted versus maximum contact pressure for [ = 1.0 and Ob =0-4y-
0.7
E.3-- yield ing analysis ------E) max cy analysis 0.5 ------
0 .4 ------
N 0.3 ------
0 .2 ------
0 0.2 0.4 0.6 0.8 1 1.2 1.4 P / a 0 Y
Figure 5-14: Optimum peening depth plotted versus maximum contact pressure for [t = 0.8, Ub=0-10Y-
73 Parameter Value AKo 3 MPafiui R 0.1 a 50 Lm Y 1
Table 5.1: Parameters and respective values for the threshold stress intensity range analysis.
5.3 Stress Intensity Range Threshold Analysis
The maximum principal stress analysis is a conservative approach. An alternative to the maximum principal analysis is an analysis based on the threshold stress intensity factor range AKo. At stress intensity factor ranges below the threshold, cracks either remain dormant or grow at undetectable rates [24]. This could be used in place of the maximum principal stress
analysis to determine the optimum peening depth. Using the values listed in Table 5.1, a MATLAB script file was written to determine the peening depth such that AKapplied < AKO. The expression used for AKapplied is given by:
AKapplied = Y(Ao)/7a (5.3)
where Y is a geometrical factor.
The analysis was performed using Ob = .6ay, and p = 0.8. The material stresses in the
x-direction were used to determine Au. Figure 5-15 shows the results of the analysis, where
the required peening depth is plotted versus the maximum contact pressure. In this figure the
results of the maximum principal stress analysis for the same loading conditions are also given
for comparison. As can be seen in the figure, the maximum principal stress analysis is the
more conservative analysis.
5.4 Protective Coatings
The protective coatings analysis in this work focuses on the ability of the coating to change the coefficient of friction of the system. Again a sphere on flat plane geometry is used in this
section. A change in the coefficient of friction has a direct impact on the contact stresses. The
contact stresses displayed in Equations 2.10 and 2.11 are shown again below in Equations 5.4
74 0.4
0 .3 ~------
0 .2 ------
N ------0.1 1------. ------
0 ------Max Principal Stress -e Delta K
-0.1 11 11 I P0 I 0.2 0.4 0.6 0.8 1 1.2 1.4 P / T 0 Y
Figure 5-15: Comparison of the required peening depths for the maximum principal stress analysis and the threshold stress intensity range analysis.
and 5.5. The stresses at the edge of contact, at the surface, are given by:
= PO (12v)± 7r , x = -Fa, and (5.4) 1 3 8
(2 Yx= P ( 3- 1) 37rvp~8 , = -Fa. (5.5)
Therefore, reducing the coefficient of friction from a value of 1.2 to 1.0 will reduce the stresses
from ax2 = 2.160PO to o- = 1.822PO and ay = 0.291PO to a- = 0.220P, assuming v = 0.3. So it can be seen that changing the coefficient of friction has a significant impact on the stresses
induced by contact and therefore, on the fretting fatigue life of the structure. Figure 5-5 also
demonstrates the benefit of reducing the coefficient of friction. It must be remembered that
the equations shown above were for global sliding conditions. However, Chivers and Gordelier
in [16] have shown that for y > 0.2, reducing ,a will also reduce stress for stick-slip conditions.
No experiments have been conducted as part of this work in which the coefficient of friction
was purposely altered to examine the actual effect on fretting fatigue life. However, Chivers
75 and Gordelier [41] performed a number of experiments for various fretting fatigue palliatives including shot peening and protective coatings and their results do support the validity of the discussion above. They examined two types of non-metallic coatings. Both were graphite-filled resins, one of the resins was a polyimide and the other was an isocyanate epoxy. The improve- ments in fretting fatigue behavior produced by the coatings were quite significant and were observed at all stress levels. Chivers and Gordelier [41],define a term called the improvement factor, If, to measure the success of the various palliatives being investigated. If is given by:
(5.6) If =- 0- - o-f where u is the run-out stress under fretting conditions with a palliative, c-f is the run-out
stress under fretting conditions and no palliative and - is the run-out stress without fretting. Chivers and Gordelier define the run-out stress as the stress at which failure does not occur
in about 108 cycles. The improvement factor values were found to be If = 0.3 for graphite
in polyimide resin and If = 0.55 for graphite in epoxy resin. For comparison purposes the
improvement factors found for shot peening were If = 0.47 for low intensity peening and
If = 0.43 for high intensity peening. It is interesting that the graphite in epoxy resin coating actually outperformed shot peening. Chivers and Gordelier also performed experiments using a copper shim and the combination of a copper shim and graphite in epoxy. The use of the
copper shim was intended to reduce the contact stresses at the surface of the material due to
fretting, which from the previous analysis, again should improve fretting fatigue behavior. The
improvement factors found were If = 0.63 with copper shim and If = 0.77 using both a copper
shim and graphite in epoxy. Flat ended cylindrical pads made of 1Cr-Mo steel were fretted against rectangular 3.5Ni-Cr-Mo-V specimens for the experiments conducted by Chivers and
Gordelier in [41]. The reduction in fatigue life due to fretting is directly related to the stresses
at the contact surface and if the stresses can be reduced the fatigue life will be improved.
76 Chapter 6
Conclusions and Future Work
6.1 Conclusions
The key findings in this work are summarized below.
" The surface stresses due to contact can be used to characterize the damaging effects of fretting on the fatigue properties of a material. The critical fretting parameters, such
as: the normal contact load, the oscillating tangential load and the coefficient of friction
are all incorporated into the expressions for the contact stresses. It was shown that for
a given bulk loading condition, the fretting fatigue life is reduced as the effective contact
stresses are increased.
" A fretting fatigue knock-down factor curve has been produced for Al 7075-T6. It shows
the approximate relationship between the effective stress due to contact and the magnitude
of the fretting fatigue knock-down factor, KF.
" It was shown that the use of a single correction factor to modify the Basquin relationship
for fretting fatigue would require extensive experimental work. The correction factor can
be expressed as a function of the effective stress due to contact for a given bulk loading
condition. However, this relationship changes as the bulk loading is changed. Therefore, families of correction factor curves would need to be developed for various bulk loadings.
The experimental work required may be tenable if the correction factor was determined
77 to be a function of the contact stresses only and not dependent on the contact geometry for a given material.
9 A failure envelope curve has been generated for Al 7075-T6. This curve shows the approximate combination of bulk and contact loading that will produce component failure.
9 The damage tolerant approach analysis has been used to bound the crack initiation and
short crack propagation stages of fretting fatigue.
9 A new method has been proposed to determine the optimum peening depth for a com- ponent that undergoes fretting using the stress analysis provided by Hamilton [14]. The method was applied to a sphere on flat surface contact geometry, but is valid for any con- tact geometry that has analytic expressions for the stress fields. The optimum peening
depth has been determined for Ti-6A1-4V for various combinations of bulk and contact
loadings, for a given coefficient of friction. This analysis was performed for four different
values of the coefficient of friction, so that the effect of coatings on peening depth could
be analyzed.
6.2 Future Work
More experimental work needs to be performed to determine a more accurate failure envelope curve. This could prove to be a very beneficial design tool, especially if it was shown to be geometry independent. Theoretical work is also needed for this curve to better understand why the given combination of contact and bulk stresses lead to component failure. This will require a thorough examination of the crack initiation stage of fretting fatigue crack growth.
The shot peening analysis provided in this work also needs experimental validation to de- termine the actual depth of mechanical stress relieving due to the contact stresses. If the compressive residual stresses due to shot peening are relieved, the damage to the surface by the shot peening process may cause a significant reduction in the fatigue strength of the material as was discussed earlier. More experimental work also needs to be done to determine the exact location of crack initiation. The pictures displayed in the previous work do not provide conclusive evidence for
78 the location of crack initiation. Interrupted tests are recommended because they could provide not only the location of crack initiation, but could also provide information on the number of cycles required for crack initiation. More analysis is also required for the predicted location of crack initiation for fretted components. The stress analysis approach does not always predict the correct location of crack initiation.
79 Appendix A
Ti-6Al-4V Yielding Analysis Program
%Ti-6A1-4V Yielding Analysis %Input the material properties
E=115.7e9;
nu=0.321; R=12.7e-3; Sy=925e6;
K=3*(1-nu^2)/(2*E); Sres=-0.85*Sy
Saxial=-.1*Sy
%Where E:Young's Modulus, nu:Poisson's Ratio, R:Radius of fretting pads, %Sy:Yield Strength, Sres:Residual Stress, Saxial:Applied bulk stress for jj=1:8
mu=0.5;
for k=1:6 P=0.20; Ry=O;
% Increase the contact load, P, for each mu and Saxial until the effective
80 % stress exceeds Sy. while Ry<1 a=(K*P*R) ^(1/3); Po=(3*P)/(2*pi*a^2); B1=Po/a; B2=mu*B1;
%Input Hamilton's Equations y=0; z(1)=0.0000001*a; x(1)=-1.05*a; for j=1:1 for i=1:110 r=x(i); A=(r^2)+(z(j)^2)-(a^2); S=((A^2)+(4*(a^2)*(z(j)^2)))^(1/2); M=((S+A)/2)^(1/2); N=((S-A)/2)^(1/2); phi=atan(a/M); G=(M^2)-(N^2)+(z(j)*M)-(a*N); H-(2*M*N)+(a*M)+(z(j)*N);
%Stresses due to normal loads Sxn=(Po/a)*((1+nu)*z(j)*phi+(1/r^2)*((0-x(i)^2/r^2)*((1-nu)*N*z(j)^2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*z(j)*a)-N*(x(i)^2)-(M*x(i)^2*z(j)*a/S))); Syn= (Po/a)*((I1+nu) *z(j) *phi+ (1/r^-2)* ((x(i)^-2/r^-2) *((1-nu) *N*z(j)^-2- ((1-2*nu) /3) *(N*S+2*A*N+a^3)-nu*M*z(j)*a)-N*(2*nu*x(i) ^2)-(O/S))); Szn=B1*(-N+(a*z(j)*M)/S); TAzxn=B1*(-z(j)*(x(i)*N/S-(x(i)*z(j)*H/(G^2+H^2))));
%Stresses due to tangential loads
Sxt=B2* (-x(i) * (nu/4+1)*phi+ (a*x(i)*M/r^4) *((3/2-2*x(i) ^2/r^2)*(S*nu-2*A*nu+ z(j) ^2)+(x(i) ^2*z(j) ^2/S)+(7*nu*r^2/4)-(2*nu*x(i) ^2)+r^2)+(x(i)*z(j)*N/r^4)*
81 ((3/2-2*x(i) ^2/r 2) *(-S/6*(1-2*nu)-A/3*(1-2*nu)-.5*(z(j)2+3*a^2))+
(a^2*x(i) ^2/S)-(nu*r^2/4)-(7*r^2/4))+(4*a^3*x(i)*z(j)/(3*r^4))* (3/2-2*x(i)^-2/r^ 2)*(1-2*nu)); Syt=B2*((-3*nu*x(i)*phi/4)+(a*x(i)*M/r^4)*((1/2-2*0/r^2)*(S*nu-2*A*nu+z(j) ^2+ r^2*nu)+(O*z(j) ^2/S)+(3*nu*r^2/4))+(x(i)*z(j)*N/r^4)*((1/2-2*0/r^2)* (-S/6*(1-2*nu)-A/3*(1-2*nu)-.5*(z(j) 2+3*a^2))+(a^2*0/S)-(3*nu*r^2/4)-(1*r^2/4))+ (4*a^3*x(i)*z(j)/(3*r^4))*(1/2-2*O/r^2)*(1-2*nu)); Szt=B2*((z(j)*x(i)*N/(2*r^2)*(1-(r^2+z(j)^-2+a^2)/S))); TAzxt=B2*((3*z(j)*phi/2)+(a*z(j)*M/r^2)*(1+x(i)^2/r^2-x(i) ^2/S)+N/r^2* (-.75*(S+2*A)+z(j) ^2-.75*a^2-.25*r^2+z(j) ^2/2*(.5-2*x(i) ^2/r^2))); %Now combine the stresses Sxtot=Sxn+Sxt+Sres+Saxial;
Sytot=Syn+Syt+Sres;
Sztot=Szn+Szt;
TAzxtot=TAzxn+TAzxt;
%Determine J2 and von Mises effective stress
J2=1/6*((Sxtot-Sytot) ^2+(Sytot-Sztot) ^2+(Sztot-Sxtot) "2)+TAzxtot^2; H2=J2^.5; Se(i,j)=3^.5*H2;
x(i+1)=x(i)+.02*a; end z(j+1)=z(j)+.025*a;
end
[Sem,i]=max(Se); xl=x/a; xl=xl';
zl=z';
zl=zl/a;
Se=Se;
%Now give x,z location of max stress
82 [Smm,zm] =max(Sem); zmm=z(zm); zma=zmm/a; xm=i(zm); xmM=x(xm); xma=xmm/a;
%Now find the stresses at x=a, and x=-a at the surface a=(K*P*R) (1/3); Po=(3*P)/(2*pi*a^2); B1=Po/a; B2=mu*B1; x2=-a;
r=x2; Sxtna=B2*(-pi*x2/2*((nu/4)+1)); Sytna=B2*(-3*pi*nu*x2/8); Sxnna=B1*((1-2*nu)*a/3); Synna=B1*((1-2*nu)*(-a/3));
Sxtona=Sxtna+Sxnna+Sres+Saxial;
Sytona=Sytna+Synna+Sres;
J2na=1/6*((Sxtona-Sytona) 2+(Sytona-0) -2+(0-Sxtona) 2)+0^ 2;
H2na=J2na^.5;
Sena=3^.5*H2na;
%Find stresses at x=a
x2=a; r=x2; Sxtpa=B2*(-pi*x2/2*((nu/4)+1)); Sytpa=B2* (-3*pi*nu*x2/8); Sxnpa=B1*((1-2*nu)*a/3); Synpa=B1*((1-2*nu)*(-a/3)); SxtoPa=Sxtpa+Sxnpa+Sres+Saxial;
83 SytoPa=Sytpa+Synpa+Sres; J2pa=1/6*((SxtoPa-SytoPa) 2+(SytoPa-0) 2+(0-SxtoPa) 2) +0 2; H2pa=J2pa^.5;
Sepa=3^.5*H2pa; %Now determine the maximum effective stress
Smmm= [Smm,Sena,Sepa]
[Smmax,xmmax]=max(Smmm);
Semax=Smmax;
Ry=Semax/Sy; P=P+.2; end P1=P-.2; P2(k,jj)=P1; mu=mu+.1;
end
Saxial=Saxial+(.1*Sy)
end
mu=mu; P2=P2 %Now put P terms over yield strength.
for n=1:8
for i=1:6 a=(K*P2(i,n)*R) (1/3); Pol(i,n)=(3*P2(i,n))/(2*pi*a^2); Ryy(i,n)=Pol(i,n)/Sy; end
end
Ryy=Ryy
84 Appendix B
Depth of Yielding Analysis
%Ti-6A1-4V Depth of Yielding Analysis %Input the material properties
E=115.7e9;
nu=0.321; R=12.7e-3; Sy=925e6;
K=3*(1-nu^2)/(2*E); Sres=-.85*Sy
mu=0.3;
Saxial=-0.1*Sy for jj=1:4 P=10; for k=1:2 a=(K*P*R) (1/3); Po=(3*P)/(2*pi*a^2); B1=Po/a;
B2=mu*B1;
%Input Hamilton's Equations
y=O; z(1)=0.0000001*a;
85 x(l)=-1.05*a; for j=1:27 for i=1:110 r=x(i); A=(r^2)+(z(j)^2)-(a^2); S=((A^2)+(4*(a^2)*(z(j)^2)))^(1/2);
M=((S+A)/2)^(1/2); N=((S-A)/2)^(1/2); phi=atan(a/M); G=(M^2)-(N^2)+(z(j)*M)-(a*N); H=(2*M*N)+(a*M)+(z(j)*N); %STRESSES DUE TO NORMAL LOADS Sxn=(Po/a) *((1+nu) *z(j) *phi+(1/r^ 2) *((O-x(i) 2/r 2) *((1-.nu) *N*z(j) 2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*z(j)*a)-N*(x(i)^2)-(M*x(i)^2*z(j)*a/S))); Syn=(Po/a) *((1+nu) *z(j) *phi+(1/r^2)*((x(i) 2/r^2) *((1-nu) *N*z(j) 2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*z(j)*a)-N*(2*nu*x(i)^2)-(0/S))); Szn=B1*(-N+(a*z(j)*M)/S); TAzxn=B1*(-z(j)*(x(i)*N/S-(x(i)*z(j)*H/(G^2+H^2))));
%Stresses due to tangential loads Sxt=B2* (-x(i)*(nu/4+1)*phi+(a*x(i) *M/r^4)*((3/2-2*x(i) 2/r^ 2)* (S*nu-2*A*nu+z(j)^2)+(x(i)^2*z(j)^2/S)+(7*nu*r^2/4)-(2*nu*x(i)^2)+r^2)+ (x(i)*z(j)*N/r^4)*((3/2-2*x(i)^2/r^2)*(-S/6*(1-2*nu)-A/3*(1-2*nu)- .5*(z(j)^2+3*a^2))+(a^2*x(i)^2/S)-(nu*r^2/4)-(7*r^2/4))+ (4*a^3*x(i)*z(j)/(3*r^4))*(3/2-2*x(i)^2/r^2)*(1-2*nu)); Syt=B2*((-3*nu*x(i)*phi/4)+(a*x(i)*M/r^4)*((1/2-2*0/r^2)* (S*nu-2*A*nu+z(j)^2+r^2*nu)+(O*z(j)^2/S)+(3*nu*r^2/4))+(x(i)*z(j)*N/r^4)* ((1/2-2*0/r^2)*(-S/6*(1-2*nu)-A/3*(1-2*nu)-.5*(z(j)^2+3*a^2))+(a^2*0/S)- (3*nu*r^2/4)-(1*r^2/4))+(4*a^3*x(i)*z(j)/(3*r^4))*(1/2-2*0/r^2)*(1-2*nu)); Szt=B2*((z(j)*x(i)*N/(2*r^2)*(M-(r^2+z(j)-2+a-2)/S))); TAzxt=B2*((3*z(j)*phi/2)+(a*z(j)*M/r^2)*(1+x(i)^2/r^2-x(i)^2/S)+N/r^2*
86 (-.75*(S+2*A)+z(j) ^2-.75*a^2-.25*r2+z(j) 2/2*(.5-2*x(i) ^2/r^2))); %Now combine the stresses
Sxtot=Sxn+Sxt+Sres+Saxial;
Sytot=Syn+Syt+Sres;
Sztot=Szn+Szt;
TAzxtot=TAzxn+TAzxt;
%Determine J2 and von Mises effective stress J2=1/6*((Sxtot-Sytot)^2+(Sytot-Sztot)^2+(Sztot-Sxtot) 2)+TAzxtot^2; H2=J2^.5; Se(i,j)=3^.5*H2; x(i+1)=x(i)+.02*a; end z(j+1)=z(j)+.02*a; end
[Sem,i] =max(Se); xl=x/a; xl=xl'; z12=z'; zl3=zl2/a;
Se=Se;
%Now give x,z location of max stress
[Smm,zm]=max(Sem); zmm=z(zm); zma=zmm/a; xm=i(zm); xmm=x(xm); xma=xmm/a;
%Now determine when von Mises effective stress equals yield stress
Ry=2; zma=zma
87 zl=zma*a; xl(1)=-1.05* a; while Ry>1 for i=1:110 a=(K*P*R) (1/3); Po-(3*P)/(2*pi*a^2); B1=Po/a;
B2=mu*B1; r=xl (i); A=(r^2)+(zl^2)-(a^2); S=((A^2)+(4*(a^2)*(zl^2))) (1/2); M=((S+A)/2)^(1/2); N=((S-A)/2)^(1/2); phi=atan(a/M); G=(M^ 2)-(N^2)+(zl*M)-(a*N); H=(2*M*N)+(a*M)+(z1*N); %Stresses due to normal loads Sxn=(Po/a)*((1+nu)*z1*phi+(1/r^2)*((0-x1(i) 2/r^2)*((l-nu)*N*z1 2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*zl*a)-N*x1(i) ^2-(M*x1(i) ^2*zl*a/S))); Syn=(Po/a)*((1+nu)*zl*phi+(1/r^2)*((xl(i) ^2/r^2)*((1-nu)*N*zl "2- ((1-2*nu) /3)* (N*S+2*A*N+a^ 3)-nu*M*zl*a)-N*(2*nu*xl (i) ^2)-(0/S))); Szn=B1*(-N+(a*zl*M)/S); TAzxn=B1*(-z1*(x1 (i)*N/S-(x1(i)*z1*H/ (G^ 2+H^ 2)))); %Stresses due to tangential loads Sxt=B2* (-x1(i)*(nu/4+1)*phi+(a*x1 (i) *M/r^4) *((3/2-2*xl (i) ^2/r^2)* (S*nu-2*A*nu+z1^ 2)+(xl(i) ^2*zl^2/S)+(7*nu*r^2/4)-(2*nu*x1(i) "2)+r"2)+ (x1(i)*zI*N/r^4)*((3/2-2*xI(i)^-2/r^2)*(-S/6*(1-2*nu)-A/3*(1-2*nu)-.5* (z1^2+3*a^2))+(a^2*x1(i)^-2/S)-(nu*r^2/4)-(7*r^2/4))+(4*a^3*xl(i)*zl/(3*r^4))* (3/2-2*x1(i) ^2/r^2)*(1-2*nu)); Syt=B2*((-3*nu*xl(i)*phi/4)+(a*xl(i)*M/r^4)*((1/2-2*0/r^2)*(S*nu-2*A*nu+zl "2+
88 r^2*nu)+(O*zl 2/S)+(3*nu*r 2/4))+(xl(i)*z1*N/r^ 4)*((1/2-2*0/r^2)*(-S/6* (1-2*nu)-A/3*(1-2*nu)-.5*(zl^ 2+3*a^2))+(a2*0/S)-(3*nu*r^2/4)-(1*r^2/4))+
(4*a^ 3*x1 (i) *zI/ (3*r^ 4))*(1/2-2*0/r ^2)*(1-2*nu)); Szt=B2*((zl*x1(i)*N/(2*r^2)*(1-(r^ 2+zl ^2+a^2)/S))); TAzxt=B2*((3*zl*phi/2)+(a*zl*M/r^2)*(1+xl(i) 2/r^2-xl(i) 2/S)+N/r^2* (-.75*(S+2*A)+zl ^2-.75*a^2-.25*r^2+z^2/2*(.5-2*x1(i)^2/r^2)));
%Now combine the stresses
Sxtot=Sxn+Sxt+Sres+Saxial; Sytot=Syn+Syt+Sres;
Sztot=Szn+Szt;
TAzxtot=TAzxn+TAzxt;
%Determine J2 and von Mises equivalent stress
J2=1/6*((Sxtot-Sytot)^2+(Sytot-Sztot) 2+(Sztot-Sxtot) 2)+TAzxtot^2; H2=J2^.5; Se1(i)=3^.5*H2; xl(i+1)=xl(i)+.02*a; end
Seml=max(Sel);
Ry=Seml/Sy zl=zl+.O1*a; end z11(k,jj)=(zl/a)-.01; P=P+40; end
Saxial=Saxial+.l*Sy end z11=z11 mu=mu;
89 Appendix C
Maximum Principal Stress Analysis
%Ti-6A1-4V Maximum Principal Stress Analysis %Input the material properties E=115.7e9; nu=0.321; R=12.7e-3; Sy=925e6;
mu=0.30; K=3*(1-nu^2)/(2*E);
Sres=-. 8 5 *Sy;
Saxial=-.1*Sy; for cc=1:8
P=10;
for k=1:11
Slmax=1; a=(K*P*R) (1/3); Po=(3*P)/(2*pi*a^ 2); B1=Po/a; B2=mu*B1; %Input Hamilton's Equations
y=0;
90 z=0.0000001*a; x(l)=-1.05*a; while Slmax>0
%Slmax is the maximum principal stress for i=1:110 r=x(i);
A=(r^2)+(z^2)-(a^2); S=((A^2)+(4*(a^2)*(z^2)))^(1/2); M=((S+A)/2)^(1/2); N=((S-A)/2)^(1/2); phi=atan(a/M); G=(M^2)-(N^2)+(z*M)-(a*N); H=(2*M*N)+(a*M)+(z*N); %STRESSES DUE TO NORMAL LOADS Sxn=(Po/a)*((1+nu)*z*phi+(1/r^2)*((O-x(i)^2/r^2)*((l-nu)*N*z^2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*z*a)-N*(x(i)^2)-(M*x(i)^2*z*a/S))); Syn=(Po/a)*((1+nu)*z*phi+(1/r^2)*((x(i)^2/r^2)*((1-nu)*N*z^2- ((1-2*nu)/3)*(N*S+2*A*N+a^3)-nu*M*z*a)-N*(2*nu*x(i)^2)-(0/S))); Szn=B1*(-N+(a*z*M)/S); TAzxn=B1*(-z*(x(i)*N/S-(x(i)*z*H/(G^ -2+H2)))); %Stresses due to tangential loads Sxt=B2*(-x(i) *(nu/4+1) *phi+(a*x(i) *M/r ^ 4) *((3/2-2*x(i) 2/r^2)* (S*nu-2*A*nu+z^2)+(x(i) 2*z^2/S)+(7*nu*r^2/4)-(2*nu*x(i) ^2)+r^2)+ (x(i)*z*N/r^4)*((3/2-2*x(i) ^2/r 2)*(-S/6*(1-2*nu)-A/3* (1-2*nu)- .5*(z ^2+3*a^ 2)) +(a^ 2*x(i) 2/S)-(nu*r 2/4)-(7*r 2/4)) + (4*a^ 3*x(i)* z/ (3*r 4)) * (3/2-2*x(i) 2/r 2)*(1-2*nu));
Syt=B2* ((-3*nu*x(i) *phi/4) + (a*x(i) *M/r 4) * ((1/2-2*0/r 2)* (S*nu-2*A*nu+z 2+r^ 2*nu) + (O*z^2/S) + (3*nu*r ^2/4)) + (x(i) *z*N/r 4)*
((1/ 2-2*0/r2)*(-S/6* (1-2*nu)-A/3*(1-2*nu)-. 5*(z ^2+3*a^ 2)) + (a^-2*0/S)- (3*nu*r^-2/4)- (1 *r^-2/4)) +(4*a^-3*x(i) *z/ (3*r^-4)) *
91 (1/2-2*0/r^ 2) *(1-2*nu)); Szt=B2*((z*x(i) *N/(2*r^2)*(1-(r^ 2+z^2+a^2)/S))); TAzxt=B2*((3*z*phi/2)+(a*z*M/r^2)*(1+x(i) 2/r^2-x(i) 2/S)+ N/r^2*(-.75*(S+2*A)+z2-.75*a^2-.25*r^2+z^2/2*(.5-2*x(i)^ 2/r^2)));
%Now combine the stresses
Sxtot=Sxn+Sxt+Sres+Saxial;
Sytot=Syn+Syt+Sres; Sztot=Szn+Szt;
TAzxtot=TAzxn+TAzxt;
Sxtotv(i) =Sxtot; Sztotv(i)=Sztot; TAzxtotv(i)=TAzxtot;
%Determine the max principal stress for each point
cxz=(Sxtot+Sztot)/2;
rxz=((Sxtot-cxz) ^2+TAzxtot^2) .5;
Slxz(i)=cxz+rxz;
x(i+1)=x(i)+.02*a;
end %Find the maximum principal stress of the vector of principal stresses Slxzm=max(Slxz);
Sxmax=max(Sxtotv);
Szmax=max(Sztotv);
TAzxmax=max(TAzxtotv);
Slmax=Slxzm
z=z+.01*a; zaa=z/a
end z2(k,cc)=z-.01*a;
a=a; z3(k,cc)=(1/a)*z2(k,cc);
92 P=P+40 end
Saxial=Saxial+. 1*Sy end z2=z2 z3=z3 mu=mu;
93 Appendix D
Hamilton's Expressions
D.1 Normal Load Stresses
2 3 3P 1 [(1 - v) Nz - 1 (NS + 2AN + a ) - vMza X = 21r + V) zO +1 [(1 (2 + 2vy2) - MX za { -N
3 3P 1 [(1 - v) NZ -1v (NS + 2AN + a ) - vMza] 3 Y 27ra 7 2 2 2 [ { -N (y + 2vx ) _ My za
3 P 3 - a z M l crz =-7r -N±+
At y = 0, the only non-zero shear stress is TzX.
3P ' N xzH Tz - 2 -z S G2+H
where
2 2 2 1 2 2 2 A = r2 + z2 - a2; S - (A + 4a z ) / ; r = 2 + y and
=(A)1/2 ;N = (S-A)1/ 2 ; # = tan-- (--) and
94 G = M 2 N 2 +zM-aN; H=2MN+aM+zN On the axis where r = 0, the formulae above are indeterminate and must be re-worked. Hamilton performed this and the expressions are now given by:
O = y 2 3 (1+ v) (z tan-' (a/z) - a)+ 2(a2 +z2)
37ra3 -- - ~ 2 2
27ra3 a2 + z 2
and Txy = Tyz = TzX = 0.
D.2 Tangential Load Stresses
-x (.L+1)0+ a- (I- 2X) (Su - 2Av + z2) + X2Z2 -2vx2 + r2} - 3Q a2X2 r2+7r2} 0X- 2ira3 +xzN {2 -(12v)- A(1-2v)-1(2+3a2)]+ 3 - 2v) [ 4a3Txz (3 2 2X2)=r/ (1 I
2 +vOaxM I - V_) [(Sv - 2Av~rv
= 3Q 2 2 3 +xzN 12 - 2V) - (1- 2v) - (Z2 + 3a )] + _ 3vr2+r2 - 27ra 3 + 4a3r4_xz (~22 } (1- 2V)2 I Again the only non-zero shear stress at y = 0 is TzX.
* Note the equation for ax in the original paper by Hamilton contained an error. The paper
contained the number 3 in the expression marked above where it should have contained an S.
The units do not work if this change is not made.
-2 (S + 2A) + Z2 3a2+r2 3Q 3z+ azM x2 X2 N +4 4 2 Tzx 2 7ra3 2 r2 +72 + _
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